Q1. if A is a subset of X, choose the topology on X as {∅,U|for every U in X that A is a subset of U}. Then is this topology a locally compact space?(adsbygoogle = window.adsbygoogle || []).push({});

Q2. X=[-1,1], the topology on X is {∅, X, [-1, b), (a,b), (a,1] | for all a<0<b}. How to prove every open set (a,b) in X is NOT locally compact?

for Q1, there is so few restrictions on X, I don't know whether it's a locally compact space or not, however, I also cannot find a counterexample.

for Q2, when defined open set of a specific topology, does is mean that we have also defined closed set? If so, then for every point a in X, a closed set including a is a compact subset, then it is locally compact, a contradiction.

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# Questions on locally compact space

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