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Recently I'm interested in this open question:

Must every star compact topological group be countably compact?

  1. star compactness ( which implies pseudocompactness ) = for every open cover $U$ of the space $X$, there exists a compact subspace $K$ such that $\cup \{u \in U: u \cap K \text{ is not empty} \} = X.$
  2. countably compact ( which implies star compactness obviously) = for every open cover $U$ of the space $X$, there exists a finite subspace $K$ such that $\cup \{u \in U: u \cap K \text{ is not empty} \} = X.$ This definition is under the $T_1$ assumed. It is equivalent to this: for every countable open cover of $X$ there is a finite subcover of $X$.

I'm not very familar with topological group. I have some questions:

Firstly, could someone complete to list the properties of star compact topological group. These of star compact topological group are I know, for example:

  1. Tychonoff = $T_0$ in every topological group,
  2. pseudocompactness from star compactness,
  3. CCC = countable chain condition, for every pseudocompact topological group has the CCC.

Secondly, if you have any idea for this open question, you could write here.

Thirdly, Is there a pseudocompact topological group but is not separable?

Thanks for any help:)

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  • $\begingroup$ How does this question arise? Can you give some examples of topological groups where you can check that they are star compact without simultaneously verifying stronger compactness properties? $\endgroup$ – t.b. Jul 1 '12 at 11:19
  • $\begingroup$ Just to contribute something mildly constructive: take $G = (\mathbb{Z}/2\mathbb{Z})^{I}$ where $I$ is a large enough set to get an example of a compact Hausdorff group that isn't separable (a separable Hausdorff space has cardinality at most $2^{\mathfrak{c}}$ as was shown in this thread for example). $\endgroup$ – t.b. Jul 1 '12 at 14:58
  • $\begingroup$ Since start compactness is perhaps less familiar, I'll add a link to another question where the definition is given. $\endgroup$ – Martin Sleziak Jul 2 '12 at 10:24
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Must every star compact topological group be countably compact?

Yes, as far as I know, this is an open problem, see [vMTW]. At the beginning of this month I recalled it at our seminar , but as professionals we are too busy and too lazy to devote a lot of time for a single problem, but you may try to attack it. Good luck!

could someone complete to list the properties of star compact topological group.

As far as I know, a little is known, because topological groups with star covering properties are too specific, so they are thought as a class intermediate between (countably) compact and pseudocompact top groups. The structural theory of compact topological groups is deeply developed (see, for instance [Pon]), but is not directly applicable to star compact groups. On the other hand, since star compact groups are pseudocompact, they are totally bounded, that is subgroups of compact groups. Similarly to countably compact and pseudocmpact groups, we can look for star compact groups in the form of groups contained between $\Sigma$-products and Tychonoff products of compact groups. I don’t know whether a product of a family star compact group is star compact. But, at least, it is pseudocompact, because a Tychonoff product of a family of (Tychonoff) pseudocompact topological groups is pseudocompact, see [CR].

Another approach is to study star covering properties of topological spaces, which is already a well investigated ground, see, for instance, surveys [Mat] and [DRRT].

References

[CR] W. W. Comfort, Kennet A. Ross Pseudocompactness an uniform continuity in topological groups, Pacif. J. Math. 16:3 (1966), 483-496.

[DRRT] E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree Star covering properties. Topology Appl, 39:1 (1991), 71-103.

[Mat] M. Matveev A Survey on Star Covering Properties.

[vMTW] J. van Mill, V.V. Tkachuk, R.G. Wilson Classes defined by stars and neighbourhood assignments, Topology Appl. 154:10 (2007), 2127-2134.

[Pon] Lev S. Pontrjagin Topological groups, Gordon and Breach, 1966.

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