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GATE 2024 MA – Question 13

Topology · Topological Spaces and Constructions · 1 mark · Multiple choice

For A⊆X in any topological space, write int, ∂ and overbar for interior, boundary and closure. Which inclusion need not hold?

  1. $\operatorname{int}(X\setminus A)\subseteq X\setminus\bar A$
  2. $A\subseteq\bar A$
  3. $\partial A\subseteq\partial(\operatorname{int}A)$
  4. $\partial(\bar A)\subseteq\partial A$

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Correct answer: (C) $\partial A\subseteq\partial(\operatorname{int}A)$

Explanation

Take A=ℚ in ℝ. Its boundary is all ℝ, but its interior is empty and the empty set has empty boundary, refuting **C**. A is the standard identity int(X∖A)=X∖closure(A); B is the closure property; D follows because int(A)⊆int(closure(A)).

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