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Let $X$ be a topological space and let $A \subseteq X$. Let $D(A)$ denote the boundary of $A$, i.e. the set of points in the closure of $A$ which are not in the interior of $A$. A closed set is nowhere dense if its interior is the empty set. Pick out the true statements:

(a) If $A$ is open, then $D(A)$ is nowhere dense.

(b) If $A$ is closed, then $D(A)$ is nowhere dense.

(c) If $A$ is any subset, then $D(A)$ is always nowhere dense.

if I consider open and closed intervals in real line both (a) and (b) are true, but what is the general case?

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Does "let A X" mean "let $A\subseteq X$"? – Jason DeVito Sep 5 '12 at 17:48
yes. and sorry for the typing error. – poton Sep 5 '12 at 17:53
up vote 1 down vote accepted

(a) is true: if $x$ is an interior point of $\bar{A} - A$, then $x$ has a neighbourhood disjoint from $A$, which is to say $x\notin \bar{A}$, a contradiction.

(b) is true: if $x$ is an interior point of $A - A^\circ$, then $x$ has a neighbourhood lying wholly in $A$, which is to say $x\in A^\circ$, a contradiction.

(c) is false: consider $\mathbf{Q}\subset\mathbf{R}$.

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(a) and (b) are closely related, since $D(A) = D(A^c)$. – Sean Eberhard Sep 5 '12 at 18:02

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