Interiors, closures and boundaries of topological spaces

Following a similar discussion for metric spaces, we could make similar notions for topological spaces. Let’s go through the usual definitions first. Given a topological space XXX with a subset S⊂XS \subset XS⊂X, the interior int(S)\text{int}(S)int(S) of SSS is defined to be the union of all open subsets of XXX that contains SSS. The closure S‾\overline{S}S is defined to be the intersection of all closed subsets of XXX containing SSS. S‾\int(S)\overline{S} \backslash \text{int}(S)S\int(S) is kno...

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