Boundary (topology)

In <a href="/facts/Topology/2EqW47cU">topology</a> and <a href="/facts/Mathematics/pxTouaz4">mathematics</a> in general, the boundary of a subset S of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X is the set of points in the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of S not belonging to the <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include 
 
 
 
 bd
 ⁡
 (
 S
 )
 ,
 fr
 ⁡
 (
 S
 )
 ,
 
 
 {\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),}
 
 and 
 
 
 
 ∂
 S
 
 
 {\displaystyle \partial S}
 
.
Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a <a href="/facts/Manifold/rXPERJtu">different definition</a> used in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a> and the theory of <a href="/facts/Manifold/rXPERJtu">manifolds</a>. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to <a href="/facts/Felix_Hausdorff/Rn6DuY2x">Hausdorff</a>'s border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.

Boundary (topology) open-in-new

Boundary (topology)