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Context: moving local properties to global properties of compact spaces.

If $X$ has this property locally, i.e, every point has a neighborhood has the property $P$ then $X$ itself has the property. In fact each such open neighborhood forms a open cover of $X$ .... ; but choosing $x_i$ appropriately.

Klaus Janich Topology

I am wondering, there can be a $x \in X$ but every open set $O \neq X$ imply $x \notin O$ then the only open cover of $X$ itself is $\{X\}$ right?

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If $X$ is equipped with anything but the trivial topology then $\{X\}$ is not the only open cover of $X$. But since $X$ is the only open set about $x$ any open cover of $X$ must contain $X$.

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