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anonymous
 4 years ago
Is \((0,1)\cup\{2\}\) open or closed?
anonymous
 4 years ago
Is \((0,1)\cup\{2\}\) open or closed?

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i would not say open because the complement is not closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0depends on your definition of open but one definition is for every element in A there is an open neighborhood contained in A that is not the case for the number 2

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0A is open if \[\forall x \in A \exists \epsilon\text { such that } (x\epsilon,x+\epsilon)\in A\]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0also clearly not closed, since it does not contain all its limit points (namely 0 and 1)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Theorem: If a set \(A\) has an isolated point, then it cannot be an open set. Proof: Let \(a\in A\) be an isolated point in A. Then there exists an \(\epsilon\)neighborhood \(V_{\epsilon}(a)\) such that \(V_{\epsilon}(a)\cap A=\{a\}\). It follows that \(A\) cannot be open since there exists no \(\epsilon\)neighborhood \(V_{\epsilon}(a)\) such that \(V_{\epsilon}(a)\subseteq A\). \(\blacksquare\)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0(with \(\epsilon>0\), that is)
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