Decomposition of Topological Space Into Interior, Boundary and Exterior
Naturally, given the way that the interior, boundary and exterior of a set is defined in topological spaces, one may assume that something cannot be simultaneously in any two of these sets, and furthermore that they cover the entire space.
That is, if we consider some topological space
and the interior of some set
then the interior of the complement will be disjoint from the interior of
and the remaining space not in either set will be their common boundary:
This result formalises that idea.
Given any topological space
Proof
This result follows very cleanly from the definitions of
if and only if there exists a neighbourhood of such that:
N \subseteq S. if and only if there exists a neighbourhood of such that:
N \subseteq S^c. if and only if for all neighbourhoods of :
N \cap S \neq \varnothing \quad \text{and} \quad N \cap S^c \neq \varnothing.
These cases are then clearly mutually exclusive, and cover all such possible values of