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A point $p$ in a topological space $X$ is said to be generic if $\overline{\{p\}} =X$ (i.e. $\{p\}$ is dense in $X$).

Let $G(X)=\{p \mid p\text{ is generic in }X\}$. That is, $G(X)$ is the set of all the points dense in $X$.

$X$ path-connected if $G(X)$ is nonempty.

Show $G(X)$ is a compact subspace of $X$.

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    If G(X) is finite I consider the smallest open sets containing each$p$in G(X) which implies a finite subcollection of open covers. But what if G(X) is infinite?2011-12-06

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There isn't much to add to Chris' hint in the comment. Since every nonempty open set contains every point of $G(X)$, given any cover of $G(X)$, every element of the cover contains all of $G(X)$, so every element forms a finite subcover of $G(X)$ by itself.

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    I was reading too much into it! Thanks guys!2011-12-06