For a subset S of a topological space, let Int(S) and S denote interior and closure. Which is true?

If S is open, then $S=\operatorname{Int}(\overline S)$.

If the boundary of S is empty, S is not closed.

If $\overline S\setminus S$ is a proper subset of the boundary of S, S is open.

GATE 2020 MA – Question 47

Topology · Topological Spaces and Constructions · 2 marks · Multiple choice

For a subset S of a topological space, let Int(S) and $\overline S$ denote interior and closure. Which is true?

If S is open, then $S=\operatorname{Int}(\overline S)$.
If the boundary of S is empty, S is open.
If the boundary of S is empty, S is not closed.
If $\overline S\setminus S$ is a proper subset of the boundary of S, S is open.

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Correct answer: (B) If the boundary of S is empty, S is open.

Explanation

Boundary is $\overline S\setminus\operatorname{Int}(S)$. If it is empty, $\operatorname{Int}(S)=\overline S$. Since the interior is contained in S and S in its closure, all three coincide, so S is both open and closed. This proves B and disproves C. Open sets need not be regular open, so A is not universal; the proper-subset condition in D does not force every point of S to be interior.

GATE 2020 MA – Question 47

Explanation

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