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In order to solve a problem I need to prove that if U and V are open sets in a metric space then $U \setminus V \neq \emptyset \iff \text{int}(U\setminus V)\neq \emptyset$, but I'm not sure if it is true. Can anyone help me??

Edit: Sorry but I got confused and really this was what I wanted to ask.

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2 Answers 2

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Suppose $U$ is the line $\mathbb R$ and $V$ is the line minus one point, say $\mathbb R\setminus\{0\}$. Then $U\setminus V=\{0\}$ is not empty, but has empty interior, hence the LR implication fails without additional assumptions. The RL implication is of course true.

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This is not true in general, If we let $U = (0,1)$ and $V = (0,.5) \cup (.5,1)$. Then $U \setminus V = \{.5\}$, which is nonempty however int$(U\setminus V) = $ int$(\{.5\}) = \emptyset$, so we have a counterexample.

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