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Can a nonclosed open subset of a $T_1$ topological space be compact? I mean an open compact set which is not clopen.

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up vote 12 down vote accepted

Another example: $[0,1]\cup\{0_2\}$, where $0_2$ is "another copy of $0$". A basis for the topology consists of all open subsets of $[0,1]$ with its standard topology, along with all sets of the form $\{0_2\}\cup(0,a)$ with $0<a<1$. (This is a modification of the line with two origins.)

In this space, $[0,1]$ compact, open, and not closed.

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is it $T_1$? .? – user59671 Feb 15 '13 at 5:26
    
(+1) Very nice example! – Asaf Karagila Feb 15 '13 at 5:26
    
@CutieKrait: Yes. Are there any singletons whose closedness isn't clear? – Jonas Meyer Feb 15 '13 at 5:27
    
Why isn't it hausdorff!? – user59671 Feb 15 '13 at 5:29
    
@CutieKrait: Because all neighborhoods of $0$ & $0_2$ meet. (Or because it has a compact nonclosed subset.) – Jonas Meyer Feb 15 '13 at 5:30

Consider cofinite topology on an infinite set, it is $T_1$ and every set is compact. In particular nonempty open sets which are not closed.

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