A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

$St(K, \mathscr{U})=\cup\{u\in \mathscr{U}: u \cap K \neq \emptyset\}‎$‎‎

‎ let‎ ‎‎$f$ ‎be ‎bijection‎ ‎continuous ‎function ‎from ‎‎‎ ‎compact ‎spce ‎$‎X‎$ ‎to ‎ ‎ ‎Hausdorff ‎space ‎$‎Y‎$‎.‎‎

We ‎know ‎that:

1: ‎the ‎closed ‎subset in compact ‎space is compact.‎‎

2: the ‎continuous ‎image ‎of ‎compact ‎set ‎is ‎compact.‎ ‎

3:the ‎compact ‎set ‎in ‎Hausdorff ‎space ‎is ‎closed.‎ ‎‎‎

‎Can the above theorems be expressed for a ‎star-‎ compact ‎space?‎‎ I mean, ‎is ‎closed ‎subset ‎in a‎ ‎star-compact ‎space ‎star-compact?‎ or ‎is ‎star-compact ‎set ‎in‎ ‎in ‎Hausdorff ‎space ‎‎ ‎closed?‎ ‎ ‎ ‎

  • $\begingroup$ $\omega_1$ with the order topology is star-compact, but not closed in $\omega_1+1$. Moreover, $\omega_1$ is star-finite, that is we can take the kernel $K$ to be a finite set. Pressing down lemmas used to find a single $\alpha$ so $St(\alpha,\mathcal U)$ contains some tail $[\beta,\omega_1)$ then use that $[0,\beta]$ is compact to find a finite set that works for it. $\endgroup$ – Mirko May 22 '19 at 12:11
  • 1
    $\begingroup$ closed set in a star-compact space is star-compact, has a trivial proof. Did you even try to think of the proofs before you post the questions? $\endgroup$ – Mirko May 22 '19 at 12:33
  • $\begingroup$ property "star-compact" used to mean star-finite (when a finite kernel $K$ could be found), but in more recent literature the definition of star-compact is as posted in the above question $\endgroup$ – Mirko May 22 '19 at 13:31
  • $\begingroup$ you probably do not mean, and should delete, the word "bijection". (There are some other typos too, you seemed to be in a rush posting the question ... ) $\endgroup$ – Mirko May 22 '19 at 13:47

Yes to 1 and 2, no to 3. I already answered 1 and 3 in the comments, re the answer of 2, one may consult the original paper where these properties were defined:

Classes defined by stars and neighbourhood assignments, J.van Mill, V.V.Tkachuk, R.G.WilsonTopology and its Applications Volume 154, Issue 10, 15 May 2007, Pages 2127-2134


In particular Proposition 3.2 there says (The proof of the following proposition is straightforwardand left to the reader):
If a class $\mathcal P$ of spaces is invariant under continuous maps, then the class of star-$\mathcal P$ spaces is also invariant under continuous maps. (In particular, the classes of star compact spaces, spaces star determined by countably compact spaces, spaces star determined by compact metrizable spaces and spaces star determined by compact countable spaces are all preserved by continuous maps.)

For the comment that $\omega_1$ is star-compact, you would need to fill in the details, which are not difficult if you know what the Pressing down lemma says. One example, how the Pressing down lemma is used in a similar context, is seen in the proof of Example 2.3 in the above paper. Try to take that proof as a model, and prove that $\omega_1$ is star-compact (this proof would be simpler than the proof of Example 2.3).


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