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The question is following:

Identify $\mathrm{Int}(A)$, $\mathrm{Bd}(A)$, A' and $\mathrm{Cl}(A)$ in the following cases:

(a) $X=\{a,b\}$ with the discrete topology, $A=\{a\}$.

My answer is the following:

  • $\mathrm{Int}(A)= \{a\}$.
  • $\mathrm{Bd}(A)=\emptyset$.
  • A'= \emptyset.
  • $\mathrm{Cl}(A)=\{a\}$

(b) $X=\{a,b\}$ with the trivial [indiscrete] topology, $A=\{a\}$.

My answer is the following:

  • $\mathrm{Int}(A)= \emptyset$.
  • $\mathrm{Bd}(A)= \emptyset$.
  • A'= \{b\}.
  • $\mathrm{Cl}(A)=\{a,b\}$.

Please correct my answer.

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    "phi" I'm guessing is meant to be $\emptyset$ (since $\phi$ looks sometimes like an empty set). $A'$ would be the derived set: the set of all limit points).2011-10-23

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If $X=\{a,b\}$ with the discrete topology, then every set is open and every set is closed. Remember that the boundary is the set of points such that every neighborhood intersects both the set and its complement.

If $A=\{a\}$, then $A$ is both open and closed, so $A$ equals its interior and its closure; therefore, the boundary is empty. The derived set is empty as well.

If the topology on $X$ is the trivial topology (the indiscrete topology), then the interior of $A=\{a\}$ is indeed empty; the closure is $X$; that means that the boundary is all of $X$ (rather than empty). The derived set is $\{b\}$.

So your only incorrect answer is the boundary of $A$ in the second problem.