It is known that every compact space is star compact ( some of authors called it star finite).
I cannot find an example for a star compact subset $\mathbb{R}$ which is not compact. Is it true that star compact subset of $\mathbb{R}$ is compact?
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http://math.stackexchange.com/questions/2131665/a-star-compact-subset-of-mathbbr-is-closed Are you friends? – Feb 06 '17 at 13:28
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I'm not sure most authors equate star compact and star finite; see the discussion under this previous Question and consider adding an explicit definition. – hardmath Feb 06 '17 at 13:32
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A space $X$ is star compact if for any open cover $\mathcal{V}$ of $X$, there is a finite $F\subseteq X$ such that $X= st(A,\mathcal{V}) = \cup{ V \in \mathcal{V}: V \cap F \neq \emptyset$ – pusing Feb 06 '17 at 13:33
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So you have no problem with my putting this in your Question, to make it self-contained? I would say this defines star finite. Note the definition given in the earlier Question for star compact. Of course $\mathbb{R}$ is a $T_4$ space, so there the notions are equivalent. – hardmath Feb 06 '17 at 13:36