Compactly generated group

<h2 id="definition">Definition</h2>
A <a href="/facts/Topological_group/cAgNMMRU">topological group</a> G is said to be compactly generated if there exists a compact subset K of G such that

⟨
        K
        ⟩
        =
        
          ⋃
          
            n
            ∈
            
              N
            
          
        
        (
        K
        ∪
        
          K
          
            −
            1
          
        
        
          )
          
            n
          
        
        =
        G
        .
      
    
    {\displaystyle \langle K\rangle =\bigcup _{n\in \mathbb {N} }(K\cup K^{-1})^{n}=G.}

So if K is symmetric, i.e. K = K −1, then

G
        =
        
          ⋃
          
            n
            ∈
            
              N
            
          
        
        
          K
          
            n
          
        
        .
      
    
    {\displaystyle G=\bigcup _{n\in \mathbb {N} }K^{n}.}

<h2 id="locally-compact-case">Locally compact case</h2>
This property is interesting in the case of <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a> topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Metric_space/gnBBRXtc">metric</a> factor groups of G. More precisely, for a sequence 

Un
of open identity neighborhoods, there exists a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> N contained in the intersection of that sequence, such that 

G/N
is locally compact metric separable (the Kakutani-Kodaira-Montgomery-Zippin theorem).

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

<ol>
<li id="fn:1">Stroppel, Markus (2006), Locally Compact Groups, European Mathematical Society, p. 44, ISBN 9783037190166. <a href="9783037190166" target="_blank">9783037190166</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Compactly generated group open-in-new

Compactly generated group