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In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this subject, can you comment?

Propoition 7: A $G$-space is compact if and only if it is simultaneously $G$-pseudocompact (it is equivalent $\beta _{G}X=\beta X$, as mentioned in the article) and $G$-Hewitt.

$G$-pseudocompact and $G$-Hewitt spaces

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    $\begingroup$ At least for me, your link does not download a paper, but rather directs me to the mathnet.ru home page (I suppose that's what it is—it's in Russian, understandably—but it certainly doesn't seem to be a particular paper). Could you give the title of the paper? $\endgroup$
    – LSpice
    Commented Nov 28 at 3:30
  • $\begingroup$ @LSpice I edited. It should be working now $\endgroup$
    – Mehmet Onat
    Commented Nov 28 at 4:34
  • $\begingroup$ Re, yes, thanks, the link now works for me; but I think that it is still helpful to know the title "$G$-pseudocompact and $G$-Hewitt spaces", so I edited it in (and changed the link to the hopefully more robust DOI while I was at it). $\endgroup$
    – LSpice
    Commented Nov 28 at 4:51

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The statement is analogous to the general result that, for Tychonoff spaces, compactness is equivalent to pseudo-compactness plus realcompactness, see the beginning of section 3.11 in Engelking's General Topology. Antonyan's definition of $G$-Hewitt is the equivariant version of the definition of realcompactness used by Engelking. And the result you are asking about is the equivariant analogue of Theorem 3.11.1. It is clear that compactness implies $G$-pseudocompactness and $G$-Hewittness. For the converse: if $X$ is $G$-pseudocompact then $\beta X$ is an extension that satisfies (b) in Antonyan's definition 2, if $X$ is also $G$-Hewitt then (a) must fail, so $X=\beta X$.

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  • $\begingroup$ First of all, thank you very much for your reply. I would like to ask another question about this. $\endgroup$
    – Mehmet Onat
    Commented Nov 28 at 11:19
  • $\begingroup$ I want to define $G$-realcompactification as follows. Let $X$ be a Tychonoff space. We will call a realcompact $G$-space $R$ as $G$-reelcompactification of $X$ satisfying the following conditions. $\endgroup$
    – Mehmet Onat
    Commented Nov 28 at 11:20
  • $\begingroup$ 1. $R$ contains $X$ as a dense invariant subspace. 2. For each $G$-equivariant map $f:X\longrightarrow Y$ into a realcompact $G$-space $Y$ can be equivariantly extended to $\overline{f}:R\longrightarrow Y $. $\endgroup$
    – Mehmet Onat
    Commented Nov 28 at 11:21
  • $\begingroup$ We will denote the maximal $G$-realcompactification of $X$ by $\upsilon _{G}X $. If $\upsilon _{G}X=X$, then $X$ will be called $G$-Hewitt. My definition should not be different from Antonyan's definition (if I am not mistaken). Based on this definition, how can we say that $X$ is compact if $\beta _{G}X=\beta X$ and $X=\upsilon _{G}X$? $\endgroup$
    – Mehmet Onat
    Commented Nov 28 at 11:22
  • $\begingroup$ @MehmetOnat In the general case $\upsilon X$ sits between $X$ and $\beta X$; so you should establish that this also holds in the equivariant case. If $X$ is $G$-pseudocompact then so is $\upsilon_GX$ and as the latter is $G$-Hewitt you would be able to conclude $\upsilon_GX=\beta_GX$. $\endgroup$
    – KP Hart
    Commented Nov 28 at 11:51

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