Let $X$ be a topological space. By definition the space $X$ is open in $X$ and so $X$ is covered by itself. This means that $X$ has a finite open cover (it is covered by only one open set which is $X$ itself), I know this is wrong because this would imply that every topological space is compact, but i don't see where is the mistake here, thank you for your help !
Definition of open cover making all spaces compact
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general-topology
1 Answers
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The definition of compact is that for every open cover, there exists a finite subcover. What you have shown is that in every space, there exists at least one open cover, namely the whole space, which has a finite subcover.
For example, the space $(1,2)$ with standard topology is not compact since we may cover the space by $\{(1,2-\frac{1}{n}): n\in\mathbb{N}\} $ but there is no finite subcover.
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0Ok I see ! I was misled by the example of the empty set, i read that the empty set is compact because it is covered by itself, although the empty set is included in any subset of $X$ I mean there are plenty of open covers of the empty set but they only considered the open cover by the empty set itself – 2017-02-16
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0but I see that the only open cover of the empty set **that is contained in** the empty set is the emptyset itself. – 2017-02-17