Subbase

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a topological space with topology 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 A subbase of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is usually defined as a subcollection 
  
    
      
        B
      
    
    {\displaystyle B}
  
 of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 satisfying one of the three following equivalent conditions:
</p>
<ol><li>
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is the smallest topology containing 
  
    
      
        B
      
    
    {\displaystyle B}
  
: any topology 
  
    
      
        
          τ
          
            ′
          
        
      
    
    {\displaystyle \tau ^{\prime }}
  
 on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 containing 
  
    
      
        B
      
    
    {\displaystyle B}
  
 must also contain 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
</li>
<li>
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of all topologies on X containing 
  
    
      
        B
        .
      
    
    {\displaystyle B.}
  
</li>
<li>The collection of open sets consisting of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and all finite <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersections</a> of elements of 
  
    
      
        B
      
    
    {\displaystyle B}
  
 forms a <a href="/facts/Basis_(topology)/8EDSDVPD">basis</a> for 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This means that every proper <a href="/facts/Open_set/QfTCIqMu">open set</a> in 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 can be written as a <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of finite intersections of elements of 
  
    
      
        B
        .
      
    
    {\displaystyle B.}
  
 Explicitly, given a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in an open set 
  
    
      
        U
        ⊊
        X
        ,
      
    
    {\displaystyle U\subsetneq X,}
  
 there are finitely many sets 
  
    
      
        
          S
          
            1
          
        
        ,
        …
        ,
        
          S
          
            n
          
        
      
    
    {\displaystyle S_{1},\ldots ,S_{n}}
  
 of 
  
    
      
        B
        ,
      
    
    {\displaystyle B,}
  
 such that the intersection of these sets contains 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and is contained in 
  
    
      
        U
        .
      
    
    {\displaystyle U.}
  
</li></ol>
<p>If we additionally assume that 
  
    
      
        B
      
    
    {\displaystyle B}
  
 <a href="/facts/Cover_(topology)/2VUdelA4">covers</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
, or if we use the <a href="/facts/Nullary_intersection/RPfUNJh9">nullary intersection</a> convention, then there is no need to include 
  
    
      
        X
      
    
    {\displaystyle X}
  
 in the third definition.
</p><p>If 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is a subbase of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
, we say that 
  
    
      
        B
      
    
    {\displaystyle B}
  
 generates the topology 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 This terminology originates from the explicit construction of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 from 
  
    
      
        B
      
    
    {\displaystyle B}
  
 using the second or third definition above.
</p><p>Elements of subbase are called subbasic (open) sets. A <a href="/facts/Cover_(topology)/2VUdelA4">cover</a> composed of subbasic sets is called a subbasic (open) cover.
</p><p>For any subcollection 
  
    
      
        S
      
    
    {\displaystyle S}
  
 of the <a href="/facts/Power_set/FVuf8PDD">power set</a> 
  
    
      
        ℘
        (
        X
        )
        ,
      
    
    {\displaystyle \wp (X),}
  
 there is a unique topology having 
  
    
      
        S
      
    
    {\displaystyle S}
  
 as a subbase; it is the intersection of all topologies on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 containing 
  
    
      
        S
      
    
    {\displaystyle S}
  
. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.
</p><p>Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set 
  
    
      
        ℘
        (
        X
        )
      
    
    {\displaystyle \wp (X)}
  
 and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the three conditions is more useful than the others.
</p>
<h3>Alternative definition</h3>
<p>Less commonly, a slightly different definition of subbase is given which requires that the subbase 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 cover 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  In this case, 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is the union of all sets contained in 
  
    
      
        
          
            B
          
        
        .
      
    
    {\displaystyle {\mathcal {B}}.}
  
 This means that there can be no confusion regarding the use of nullary intersections in the definition.
</p><p>However, this definition is not always equivalent to the three definitions above. There exist topological spaces 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 with subcollections 
  
    
      
        
          
            B
          
        
        ⊆
        τ
      
    
    {\displaystyle {\mathcal {B}}\subseteq \tau }
  
 of the topology such that 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is the smallest topology containing 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
, yet 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 does not cover 
  
    
      
        X
      
    
    {\displaystyle X}
  
. For example, consider a topological space 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 with 
  
    
      
        τ
        =
        {
        ∅
        ,
        {
        p
        }
        ,
        X
        }
      
    
    {\displaystyle \tau =\{\varnothing ,\{p\},X\}}
  
 and 
  
    
      
        
          
            B
          
        
        =
        {
        {
        p
        }
        }
      
    
    {\displaystyle {\mathcal {B}}=\{\{p\}\}}
  
 for some 
  
    
      
        p
        ∈
        X
        .
      
    
    {\displaystyle p\in X.}
  
 Clearly, 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 is a subbase of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
, yet 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 doesn't cover 
  
    
      
        X
      
    
    {\displaystyle X}
  
 as long as 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has at least 
  
    
      
        2
      
    
    {\displaystyle 2}
  
 elements. In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the <a href="/facts/T1_separation_axiom/K4exfuf7">T1 separation axiom</a> must be a cover of that space.
</p>
<h2 id="examples">Examples</h2>
<p>The topology generated by any subset 
  
    
      
        
          
            S
          
        
        ⊆
        {
        ∅
        ,
        X
        }
      
    
    {\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}
  
 (including by the empty set 
  
    
      
        
          
            S
          
        
        :=
        ∅
      
    
    {\displaystyle {\mathcal {S}}:=\varnothing }
  
) is equal to the trivial topology 
  
    
      
        {
        ∅
        ,
        X
        }
        .
      
    
    {\displaystyle \{\varnothing ,X\}.}

</p><p>If 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is a topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 is a basis for 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 then the topology generated by 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 is 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 Thus any basis 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 for a topology 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is also a subbasis for 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 
If 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 is any subset of 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 then the topology generated by 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 will be a subset of 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}

</p><p>The usual topology on the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 has a subbase consisting of all <a href="/facts/Semi-infinite/0c13Cg1u">semi-infinite</a> open intervals either of the form 
  
    
      
        (
        −
        ∞
        ,
        a
        )
      
    
    {\displaystyle (-\infty ,a)}
  
 or 
  
    
      
        (
        b
        ,
        ∞
        )
        ,
      
    
    {\displaystyle (b,\infty ),}
  
 where 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 are real numbers. Together, these generate the usual topology, since the intersections 
  
    
      
        (
        a
        ,
        b
        )
        =
        (
        −
        ∞
        ,
        b
        )
        ∩
        (
        a
        ,
        ∞
        )
      
    
    {\displaystyle (a,b)=(-\infty ,b)\cap (a,\infty )}
  
 for 
  
    
      
        a
        ≤
        b
      
    
    {\displaystyle a\leq b}
  
 generate the usual topology. A second subbase is formed by taking the subfamily where 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 are <a href="/facts/Rational_number/VJQmyY05">rational</a>. The second subbase generates the usual topology as well, since the open intervals 
  
    
      
        (
        a
        ,
        b
        )
      
    
    {\displaystyle (a,b)}
  
 with 
  
    
      
        a
        ,
      
    
    {\displaystyle a,}
  
 
  
    
      
        b
      
    
    {\displaystyle b}
  
 rational, are a basis for the usual Euclidean topology.
</p><p>The subbase consisting of all semi-infinite open intervals of the form 
  
    
      
        (
        −
        ∞
        ,
        a
        )
      
    
    {\displaystyle (-\infty ,a)}
  
 alone, where 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is a real number, does not generate the usual topology. The resulting topology does not satisfy the <a href="/facts/T1_space/K4exfuf7">T1 separation axiom</a>, since if 
  
    
      
        a
        <
        b
      
    
    {\displaystyle a<b}
  
 every <a href="/facts/Open_set/QfTCIqMu">open set</a> containing 
  
    
      
        b
      
    
    {\displaystyle b}
  
 also contains 
  
    
      
        a
        .
      
    
    {\displaystyle a.}

</p><p>The <a href="/facts/Initial_topology/bNRdJM81">initial topology</a> on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 defined by a family of functions 
  
    
      
        
          f
          
            i
          
        
        :
        X
        →
        
          Y
          
            i
          
        
        ,
      
    
    {\displaystyle f_{i}:X\to Y_{i},}
  
 where each 
  
    
      
        
          Y
          
            i
          
        
      
    
    {\displaystyle Y_{i}}
  
 has a topology, is the coarsest topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that each 
  
    
      
        
          f
          
            i
          
        
      
    
    {\displaystyle f_{i}}
  
 is <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a>. Because continuity can be defined in terms of the <a href="/facts/Inverse_image/KANefMAl">inverse images</a> of open sets, this means that the initial topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is given by taking all 
  
    
      
        
          f
          
            i
          
          
            −
            1
          
        
        (
        U
        )
        ,
      
    
    {\displaystyle f_{i}^{-1}(U),}

where 
  
    
      
        U
      
    
    {\displaystyle U}
  
 ranges over all open subsets of 
  
    
      
        
          Y
          
            i
          
        
        ,
      
    
    {\displaystyle Y_{i},}
  
 as a subbasis.
</p><p>Two important special cases of the initial topology are the <a href="/facts/Product_topology/0hg02IaX">product topology</a>, where the family of functions is the set of projections from the product to each factor, and the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>, where the family consists of just one function, the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a>.
</p><p>The <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a> on the space of continuous functions from 
  
    
      
        X
      
    
    {\displaystyle X}
  
 to 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 has for a subbase the set of functions

V
        (
        K
        ,
        U
        )
        =
        {
        f
        :
        X
        →
        Y
        ∣
        f
        (
        K
        )
        ⊆
        U
        }
      
    
    {\displaystyle V(K,U)=\{f:X\to Y\mid f(K)\subseteq U\}}

where 
  
    
      
        K
        ⊆
        X
      
    
    {\displaystyle K\subseteq X}
  
 is <a href="/facts/Compact_space/d0cgXJH7">compact</a> and 
  
    
      
        U
      
    
    {\displaystyle U}
  
 is an open subset of 
  
    
      
        Y
        .
      
    
    {\displaystyle Y.}

</p><p>Suppose that 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 is a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> topological space with 
  
    
      
        X
      
    
    {\displaystyle X}
  
 containing two or more elements (for example, 
  
    
      
        X
        =
        
          R
        
      
    
    {\displaystyle X=\mathbb {R} }
  
 with the <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a>). Let 
  
    
      
        Y
        ∈
        τ
      
    
    {\displaystyle Y\in \tau }
  
 be any non-empty open subset of 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 (for example, 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 could be a non-empty bounded open interval in 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
) and let 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 denote the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> on 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 that 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 inherits from 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 (so 
  
    
      
        ν
        ⊆
        τ
      
    
    {\displaystyle \nu \subseteq \tau }
  
). Then the topology generated by 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is equal to the union 
  
    
      
        {
        X
        }
        ∪
        ν
      
    
    {\displaystyle \{X\}\cup \nu }
  
 (see the footnote for an explanation),
<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
where 
  
    
      
        {
        X
        }
        ∪
        ν
        ⊆
        τ
      
    
    {\displaystyle \{X\}\cup \nu \subseteq \tau }
  
 (since 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 is Hausdorff, equality will hold if and only if 
  
    
      
        Y
        =
        X
      
    
    {\displaystyle Y=X}
  
). Note that if 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a <a href="/facts/Proper_subset/SJGERmbA">proper subset</a> of 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 then 
  
    
      
        {
        X
        }
        ∪
        ν
      
    
    {\displaystyle \{X\}\cup \nu }
  
 is the smallest topology <i>on 
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> containing 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 yet 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 does not cover 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (that is, the union 
  
    
      
        
          ⋃
          
            V
            ∈
            ν
          
        
        V
        =
        Y
      
    
    {\displaystyle \bigcup _{V\in \nu }V=Y}
  
 is a proper subset of 
  
    
      
        X
      
    
    {\displaystyle X}
  
).
</p>
<h2 id="results-using-subbases">Results using subbases</h2>
<p>One nice fact about subbases is that <a href="/facts/Continuity_(topology)/sKbl02pB">continuity</a> of a function need only be checked on a subbase of the range. That is, if 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 is a map between topological spaces and if 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 is a subbase for 
  
    
      
        Y
        ,
      
    
    {\displaystyle Y,}
  
 then 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 is continuous <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> 
  
    
      
        
          f
          
            −
            1
          
        
        (
        B
        )
      
    
    {\displaystyle f^{-1}(B)}
  
 is open in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 for every 
  
    
      
        B
        ∈
        
          
            B
          
        
        .
      
    
    {\displaystyle B\in {\mathcal {B}}.}
  
 
A <a href="/facts/Net_(mathematics)/msmsvMTT">net</a> (or <a href="/facts/Sequence/gV2S4uqp">sequence</a>) 
  
    
      
        
          x
          
            ∙
          
        
        =
        
          
            (
            
              x
              
                i
              
            
            )
          
          
            i
            ∈
            I
          
        
      
    
    {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
  
 converges to a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 if and only if every subbasic neighborhood of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 contains all 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 for sufficiently large 
  
    
      
        i
        ∈
        I
        .
      
    
    {\displaystyle i\in I.}

</p>
<h3>Alexander subbase theorem</h3>
<p>The Alexander Subbase Theorem is a significant result concerning subbases that is due to <a href="/facts/James_Waddell_Alexander_II/FMPYbH93">James Waddell Alexander II</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The corresponding result for basic (rather than subbasic) open covers is much easier to prove.
</p>
Alexander subbase theorem:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Let 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 be a topological space, and 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 be a subbase of 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 If every cover of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 by elements from 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 has a finite subcover, then 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is <a href="/facts/Compact_space/d0cgXJH7">compact</a>.
<p>The converse to this theorem also holds (because every cover of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 by elements of 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 is an open cover of 
  
    
      
        X
      
    
    {\displaystyle X}
  
) 
</p>
Let 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 be a topological space, and 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 be a subbase of 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is compact, then every cover of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 by elements from 
  
    
      
        
          
            S
          
        
      
    
    {\displaystyle {\mathcal {S}}}
  
 has a finite subcover.

<table><tbody><tr><th>Proof</th></tr><tr><td><p>Suppose for the sake of contradiction that the space                     X              {\displaystyle X}   is not compact (so                     X              {\displaystyle X}   is an infinite set), yet every subbasic cover from                                           S                                {\displaystyle {\mathcal {S}}}   has a finite subcover. Let                               S                      {\displaystyle \mathbb {S} }   denote the set of all open covers of                     X              {\displaystyle X}   that do not have any finite subcover of                     X        .              {\displaystyle X.}   Partially order                               S                      {\displaystyle \mathbb {S} }   by subset inclusion and use <a href="/facts/Zorn%27s_Lemma/ssCublFT">Zorn's Lemma</a> to find an element                                           C                          ∈                  S                      {\displaystyle {\mathcal {C}}\in \mathbb {S} }   that is a maximal element of                               S                .              {\displaystyle \mathbb {S} .}   Observe that:</p><ol><li>Since                                           C                          ∈                  S                ,              {\displaystyle {\mathcal {C}}\in \mathbb {S} ,}   by definition of                               S                ,              {\displaystyle \mathbb {S} ,}                                             C                                {\displaystyle {\mathcal {C}}}   is an open cover of                     X              {\displaystyle X}   and there does not exist any finite subset of                                           C                                {\displaystyle {\mathcal {C}}}   that covers                     X              {\displaystyle X}   (so in particular,                                           C                                {\displaystyle {\mathcal {C}}}   is infinite).</li><li>The maximality of                                           C                                {\displaystyle {\mathcal {C}}}   in                               S                      {\displaystyle \mathbb {S} }   implies that if                     V              {\displaystyle V}   is an open set of                     X              {\displaystyle X}   such that                     V        ∉                              C                                {\displaystyle V\not \in {\mathcal {C}}}   then                                            C                          ∪        {        V        }              {\displaystyle {\mathcal {C}}\cup \{V\}}   has a finite subcover, which must necessarily be of the form                     {        V        }        ∪                                            C                                            V                                {\displaystyle \{V\}\cup {\mathcal {C}}_{V}}   for some finite subset                                                         C                                            V                                {\displaystyle {\mathcal {C}}_{V}}   of                                           C                                {\displaystyle {\mathcal {C}}}   (this finite subset depends on the choice of                     V              {\displaystyle V}  ).</li></ol><p>We will begin by showing that                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   is not a cover of                     X        .              {\displaystyle X.}   Suppose that                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   was a cover of                     X        ,              {\displaystyle X,}   which in particular implies that                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   is a cover of                     X              {\displaystyle X}   by elements of                                           S                          .              {\displaystyle {\mathcal {S}}.}   The theorem's hypothesis on                                           S                                {\displaystyle {\mathcal {S}}}   implies that there exists a finite subset of                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   that covers                     X        ,              {\displaystyle X,}   which would simultaneously also be a finite subcover of                     X              {\displaystyle X}   by elements of                                           C                                {\displaystyle {\mathcal {C}}}   (since                                           C                          ∩                              S                          ⊆                              C                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}\subseteq {\mathcal {C}}}  ). But this contradicts                                           C                          ∈                  S                ,              {\displaystyle {\mathcal {C}}\in \mathbb {S} ,}   which proves that                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   does not cover                     X        .              {\displaystyle X.}  </p><p>Since                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   does not cover                     X        ,              {\displaystyle X,}   there exists some                     x        ∈        X              {\displaystyle x\in X}   that is not covered by                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}   (that is,                     x              {\displaystyle x}   is not contained in any element of                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}  ). But since                                           C                                {\displaystyle {\mathcal {C}}}   does cover                     X        ,              {\displaystyle X,}   there also exists some                     U        ∈                              C                                {\displaystyle U\in {\mathcal {C}}}   such that                     x        ∈        U        .              {\displaystyle x\in U.}   It follows that                     U        ≠        X              {\displaystyle U\neq X}  , because otherwise it would imply                                           C                                {\displaystyle {\mathcal {C}}}   has a finite subcover of                     X              {\displaystyle X}  , namely the subcover                     {        U        }        =        {        X        }        ,              {\displaystyle \{U\}=\{X\},}   contradicting                                           C                          ∈                  S                .              {\displaystyle {\mathcal {C}}\in \mathbb {S} .}   Since                     U        ≠        X        ,              {\displaystyle U\neq X,}   and                                           S                                {\displaystyle {\mathcal {S}}}   is a subbasis generating                     X              {\displaystyle X}  's topology (together with                     X              {\displaystyle X}  ), from the definition of the topology generated by                                           S                          ,              {\displaystyle {\mathcal {S}},}   there must exist a finite collection of subbasic open sets                               S                      1                          ,        …        ,                  S                      n                          ∈                              S                                {\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {S}}}   with                     n        ≥        1              {\displaystyle n\geq 1}   such that                    x        ∈                  S                      1                          ∩        ⋯        ∩                  S                      n                          ⊆        U        .              {\displaystyle x\in S_{1}\cap \cdots \cap S_{n}\subseteq U.}  </p><p>We will now show by contradiction that                               S                      i                          ∉                              C                                {\displaystyle S_{i}\not \in {\mathcal {C}}}   for every                     i        =        1        ,        …        ,        n        .              {\displaystyle i=1,\ldots ,n.}   If                     i              {\displaystyle i}   was such that                               S                      i                          ∈                              C                          ,              {\displaystyle S_{i}\in {\mathcal {C}},}   then also                               S                      i                          ∈                              C                          ∩                              S                                {\displaystyle S_{i}\in {\mathcal {C}}\cap {\mathcal {S}}}   so the fact that                     x        ∈                  S                      i                                {\displaystyle x\in S_{i}}   would then imply that                     x              {\displaystyle x}   is covered by                                           C                          ∩                              S                          ,              {\displaystyle {\mathcal {C}}\cap {\mathcal {S}},}   which contradicts how                     x              {\displaystyle x}   was chosen (recall that                     x              {\displaystyle x}   was chosen specifically so that it was not covered by                                           C                          ∩                              S                                {\displaystyle {\mathcal {C}}\cap {\mathcal {S}}}  ).</p><p>As mentioned earlier, the maximality of                                           C                                {\displaystyle {\mathcal {C}}}   in                               S                      {\displaystyle \mathbb {S} }   implies that for every                     i        =        1        ,        …        ,        n        ,              {\displaystyle i=1,\ldots ,n,}   there exists a finite subset                                                         C                                                          S                              i                                                          {\displaystyle {\mathcal {C}}_{S_{i}}}   of                                           C                                {\displaystyle {\mathcal {C}}}   such that                              {                      S                          i                                }                ∪                                            C                                                          S                              i                                                          {\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{S_{i}}}   forms a finite cover of                     X        .              {\displaystyle X.}   Define                                                        C                                            F                          :=                                            C                                                          S                              1                                                    ∪        ⋯        ∪                                            C                                                          S                              n                                                    ,              {\displaystyle {\mathcal {C}}_{F}:={\mathcal {C}}_{S_{1}}\cup \cdots \cup {\mathcal {C}}_{S_{n}},}  which is a finite subset of                                           C                          .              {\displaystyle {\mathcal {C}}.}   Observe that for every                     i        =        1        ,        …        ,        n        ,              {\displaystyle i=1,\ldots ,n,}                                 {                      S                          i                                }                ∪                                            C                                            F                                {\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}   is a finite cover of                     X              {\displaystyle X}   so let us replace every                                                         C                                                          S                              i                                                          {\displaystyle {\mathcal {C}}_{S_{i}}}   with                                                         C                                            F                          .              {\displaystyle {\mathcal {C}}_{F}.}  </p><p>Let                     ∪                                            C                                            F                                {\displaystyle \cup {\mathcal {C}}_{F}}   denote the union of all sets in                                                         C                                            F                                {\displaystyle {\mathcal {C}}_{F}}   (which is an open subset of                     X              {\displaystyle X}  ) and let                     Z              {\displaystyle Z}   denote the complement of                     ∪                                            C                                            F                                {\displaystyle \cup {\mathcal {C}}_{F}}   in                     X        .              {\displaystyle X.}   Observe that for any subset                     A        ⊆        X        ,              {\displaystyle A\subseteq X,}                       {        A        }        ∪                                            C                                            F                                {\displaystyle \{A\}\cup {\mathcal {C}}_{F}}   covers                     X              {\displaystyle X}   if and only if                     Z        ⊆        A        .              {\displaystyle Z\subseteq A.}   In particular, for every                     i        =        1        ,        …        ,        n        ,              {\displaystyle i=1,\ldots ,n,}   the fact that                               {                      S                          i                                }                ∪                                            C                                            F                                {\displaystyle \left\{S_{i}\right\}\cup {\mathcal {C}}_{F}}   covers                     X              {\displaystyle X}   implies that                     Z        ⊆                  S                      i                          .              {\displaystyle Z\subseteq S_{i}.}   Since                     i              {\displaystyle i}   was arbitrary, we have                     Z        ⊆                  S                      1                          ∩        ⋯        ∩                  S                      n                          .              {\displaystyle Z\subseteq S_{1}\cap \cdots \cap S_{n}.}   Recalling that                               S                      1                          ∩        ⋯        ∩                  S                      n                          ⊆        U        ,              {\displaystyle S_{1}\cap \cdots \cap S_{n}\subseteq U,}   we thus have                     Z        ⊆        U        ,              {\displaystyle Z\subseteq U,}   which is equivalent to                     {        U        }        ∪                                            C                                            F                                {\displaystyle \{U\}\cup {\mathcal {C}}_{F}}   being a cover of                     X        .              {\displaystyle X.}   Moreover,                     {        U        }        ∪                                            C                                            F                                {\displaystyle \{U\}\cup {\mathcal {C}}_{F}}   is a finite cover of                     X              {\displaystyle X}   with                     {        U        }        ∪                                            C                                            F                          ⊆                              C                          .              {\displaystyle \{U\}\cup {\mathcal {C}}_{F}\subseteq {\mathcal {C}}.}   Thus                                           C                                {\displaystyle {\mathcal {C}}}   has a finite subcover of                     X        ,              {\displaystyle X,}   which contradicts the fact that                                           C                          ∈                  S                .              {\displaystyle {\mathcal {C}}\in \mathbb {S} .}   Therefore, the original assumption that                     X              {\displaystyle X}   is not compact must be wrong, which proves that                     X              {\displaystyle X}   is compact.                     ◼              {\displaystyle \blacksquare }  </p></td></tr></tbody></table>
<p>Although this proof makes use of <a href="/facts/Zorn%27s_Lemma/ssCublFT">Zorn's Lemma</a>, the proof does not need the full strength of choice. 
Instead, it relies on the intermediate <a href="/facts/Ultrafilter_principle/KUpERdqN">Ultrafilter principle</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>Using this theorem with the subbase for 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 above, one can give a very easy proof that bounded closed intervals in 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 are compact.  
More generally, <a href="/facts/Tychonoff%27s_theorem/PhVArhKr">Tychonoff's theorem</a>, which states that the product of non-empty compact spaces is compact, has a short proof if the Alexander Subbase Theorem is used.
</p>

<table><tbody><tr><th>Proof</th></tr><tr><td><p>The product topology on                               ∏                      i                                    X                      i                                {\displaystyle \prod _{i}X_{i}}   has, by definition, a subbase consisting of <i>cylinder</i> sets that are the inverse projections of an open set in one factor.  Given a subbasic family                     C              {\displaystyle C}   of the product that does not have a finite subcover, we can partition                     C        =                  ∪                      i                                    C                      i                                {\displaystyle C=\cup _{i}C_{i}}   into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space.  By assumption, if                               C                      i                          ≠        ∅              {\displaystyle C_{i}\neq \varnothing }   then                               C                      i                                {\displaystyle C_{i}}   does not have a finite subcover.  Being cylinder sets, this means their projections onto                               X                      i                                {\displaystyle X_{i}}   have no finite subcover, and since each                               X                      i                                {\displaystyle X_{i}}   is compact, we can find a point                               x                      i                          ∈                  X                      i                                {\displaystyle x_{i}\in X_{i}}   that is not covered by the projections of                               C                      i                                {\displaystyle C_{i}}   onto                               X                      i                          .              {\displaystyle X_{i}.}   But then                                           (                          x                              i                                      )                                i                          ∈                  ∏                      i                                    X                      i                                {\displaystyle \left(x_{i}\right)_{i}\in \prod _{i}X_{i}}   is not covered by                     C        .              {\displaystyle C.}                       ◼              {\displaystyle \blacksquare }  </p><p>Note, that in the last step we implicitly used the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> (which is actually equivalent to <a href="/facts/Zorn%27s_lemma/ssCublFT">Zorn's lemma</a>) to ensure the existence of                                           (                          x                              i                                      )                                i                          .              {\displaystyle \left(x_{i}\right)_{i}.}  </p></td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Base_(topology)/8EDSDVPD">Base (topology)</a> – Collection of open sets used to define a topology</li></ul>
<h2 id="notes">Notes</h2>

<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1989) [1966]. <a href="https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w"><i>General Topology: Chapters 1–4</i></a> [<i><a href="/facts/Topologie_G%C3%A9n%C3%A9rale/qRJhs9nr">Topologie Générale</a></i>]. <a href="/facts/%C3%89l%C3%A9ments_de_math%C3%A9matique/8TQroWxE">Éléments de mathématique</a>. Berlin New York: Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-64241-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/18588129">18588129</a>.</li>
<li><a href="/facts/James_Dugundji/xCy4DgGy">Dugundji, James</a> (1966). <i>Topology</i>. Boston: Allyn and Bacon. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-697-06889-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/395340485">395340485</a>.</li>
<li><a href="/facts/James_Munkres/Ybfa6fs4">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). <a href="/facts/Upper_Saddle_River,_NJ/Ps08aEYT">Upper Saddle River, NJ</a>: <a href="/facts/Prentice_Hall,_Inc/bynWAuat">Prentice Hall, Inc</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-181629-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/42683260">42683260</a>.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science/Engineering/Math/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li>
<li>Willard, Stephen (2004) [1970]. <a href="https://books.google.com/books?id=-o8xJQ7Ag2cC"><i>General Topology</i></a>. <a href="/facts/Mineola,_N.Y./GWYvEokQ">Mineola, N.Y.</a>: <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-43479-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/115240">115240</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Rudin 1991, p. 392 Appendix A2. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Rudin's definition is less general than ours, because it effectively requires that 
  
    
      
        B
      
    
    {\displaystyle B}
  
 covers 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (see "Alternative definition" subsection below). We drop this requirement here, and assume that 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is any subset of 
  
    
      
        
          
            P
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {P}}(X)}
 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Munkres 2000, pp. 82. - Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. <a href="https://search.worldcat.org/oclc/42683260" target="_blank">https://search.worldcat.org/oclc/42683260</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Since 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is a topology on 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is an open subset of 
  
    
      
        (
        X
        ,
        τ
        )
        ,
      
    
    {\displaystyle (X,\tau ),}
  
, it is easy to verify that 
  
    
      
        {
        X
        }
        ∪
        ν
      
    
    {\displaystyle \{X\}\cup \nu }
  
 is a topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. In particular, 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is closed under unions and finite intersections because 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is. But since 
  
    
      
        X
        ∉
        ν
      
    
    {\displaystyle X\not \in \nu }
  
, 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is not a topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 an 
  
    
      
        {
        X
        }
        ∪
        ν
      
    
    {\displaystyle \{X\}\cup \nu }
  
 is clearly the smallest topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 containing 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
). <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Muger, Michael (2020). Topology for the Working Mathematician. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Muger, Michael (2020). Topology for the Working Mathematician. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Rudin 1991, p. 392 Appendix A2. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Muger, Michael (2020). Topology for the Working Mathematician. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Subbase open-in-new

Subbase