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If [tex]A\subseteq B[/tex] are both subsets of a topological space [tex](X,\tau)[/tex], is it true that any closed subset of A is also a closed subset of B?
logarithmic said:If [tex]A\subseteq B[/tex] are both subsets of a topological space [tex](X,\tau)[/tex], is it true that any closed subset of A is also a closed subset of B?
Hmm, I'm not sure. I never said A was closed. A is in B which is in X. And some other set in A, maybe call it U, is closed. I think the answer would be U is closed in X but I don't quite see why.Hurkyl said:Note that A is a closed subset of A...
Ahh i see. Thanks. But is that the only problem here? If we insist that A is compact in B, then that fixes the problem and the statement is true, right?HallsofIvy said:Hurkyl's point is that any topological space is both open and closed as a subset of itself. If A is NOT closed as a subset of topological space B, since it IS closed as a subset of itself, the statement "any closed subset of A is also a closed subset of B" is false.