Boundary (topology)

In Topology , the boundary of a subset ''S'' of a Topological Space ''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More formally, it is the set of points in the Closure of ''S'', not belonging to the Interior of ''S''. An element of the boundary of ''S'' is called a '''boundary point''' of ''S''. Notations used for boundary of a set ''S'' include bd(''S''), fr(''S''), and

\partial S

.

There are several common (and equivalent) definitions to the boundary of ''S'':

the intersection of the closure of ''S'' with the closure of its Complement :

\partial S = \bar{S} \bigcap \overline{ (X \setminus S)}.

the closure of ''S'' without the interior of ''S'': $\partial S = \bar{S}\setminus S^o$ .

a point ''p'' in ''X'' is a boundary point of ''S'' if every Neighborhood of ''p'' contains at least one point of ''S'' and at least one point not in ''S''. The boundary of ''S'' is the set of all boundary points of ''S''.

EXAMPLES

Consider the real line R with the usual topology (i.e. the topology whose Basis Sets are Open Interval s). One has

$\partial (0,5) = \partial = \partial (0,5 = \{0,5 \}\,$

$\partial \emptyset = \emptyset$

$\partial \mathbb{Q} = \mathbb{R}$

$\partial \big(\mathbb{Q}\cap\left[0,1
ight]\big) = \left[0,1
ight]$

These last two examples illustrate the fact that the boundary of a Dense Set with empty interior is its closure.

One should keep in mind the boundary of a set is a Topological notion, therefore, one changes the topology, the set boundary may change. For example, given the usual topology on R², and the closed disk
:Ω={(''x'', ''y''): ''x''²+''y''² ≤ 1},

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Boundary (topology)