star-compact space 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 $X = \operatorname{St}(K,\mathscr{U})$.

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

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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?‎
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 A: 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 
https://www.sciencedirect.com/science/article/pii/S0166864107000193
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). 
