Timeline for A question about G-Hewitt spaces
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3 at 12:04 | vote | accept | Mehmet Onat | ||
| Nov 29 at 11:51 | comment | added | Mehmet Onat | Thank you very much for your time and patience. | |
| Nov 29 at 9:31 | comment | added | KP Hart | We were working under the assumption that $X$ is $G$-pseudocompact, so if $f:X\to C(G)$ is equivariant and continuous then $f[X]$ has compact closure; because $X$ is dense in $\upsilon_GX$ the latter has this property too, so all (bounded) equivariant continuous maps from $\upsilon_GX$ can be extended to $\beta_GX$, that is, we have condition (b), because $\upsilon_GX$ is $G$-Hewitt condition (a) must fail, so $\upsilon_GX=\beta_GX$. | |
| Nov 28 at 18:24 | comment | added | Mehmet Onat | I have already looked at the article you mentioned and I will read it more carefully. But unfortunately I still cannot see why $\beta _{G}\left( \upsilon _{G}X\right) =\beta \left( \upsilon _{G}X\right) =\beta X=\beta _{G}X$ implies $\upsilon _{G}X=\beta _{G}X$ | |
| Nov 28 at 17:28 | comment | added | KP Hart | I recommend this paper by Jan de Vries, G-spaces: compactifications and pseudocompactness. The first part surveys methods of obtaining G-compactifications. The idea is to replace $\mathbb{R}$ by the space $C(G)$. When building $\beta_GX$ you use continuous functions $f:X\to C(G)$ whose image has compact closure. For building $\upsilon_GX$ you need a metric co-domain, so you look at the bounded functions only (finite distance from the zero-function); if you follow the product construction you do not necessarily get a compact closure. | |
| Nov 28 at 16:40 | comment | added | Mehmet Onat | To summarize, I don't understand why the maximal equivariant compactification $\beta_G X$ is defined by the equivariant version of the usual Stone-Čech compactification, while the $G$-Hewitt realcompactification is defined using $C(G)$ and the concept of bounded function $\varphi :X\longrightarrow C\left( G\right) $. | |
| Nov 28 at 16:36 | comment | added | Mehmet Onat | This is the case (c) of Theorem 4 in Antonyan's article. In a previous correspondence with Antonyan I asked him if condition (b) of Theorem 4 could be removed and he said that it could. | |
| Nov 28 at 16:30 | comment | added | Mehmet Onat | I think $\beta X$ does not apply to condition 2 in my definition because it is not bounded. Perhaps there is a reason why Antonyan used the bounded and continuous $\varphi :X\longrightarrow C\left( G\right) $ instead of the 2 condition in my definition. I don't quite understand why he would do such a thing instead of the equivariant version of Engelking's classical definition 3.11. | |
| Nov 28 at 16:25 | comment | added | Mehmet Onat | I think only $\beta _{G}\left( \upsilon _{G}X\right) =\beta \left( \upsilon _{G}X\right)=\beta X=\beta _{G}X$ equations do not imply that $\upsilon _{G}X=\beta _{G}X$. It seems that other arguments are needed for this. Further, I think Antonyan's definition is more restrictive than mine. The reason why you can apply $\beta X$ to (b) in Antonyan's definition 2 is that $\varphi $ is a continuous equivariant and bounded mapping. | |
| Nov 28 at 13:22 | comment | added | KP Hart | Your definition may not require it but I think you can prove that you can identify $\upsilon_GX$ as a space between $X$ and $\beta_GX$: in Engelking's book $\upsilon X$ is constructed (Theorem 3.11.10) as a subspace of $\beta X$. Once you have done that my previous comment applies. And in that case $\upsilon_GX=\beta_GX$ because $\beta_G(\upsilon_GX)=\beta(\upsilon_GX)$ (as $\upsilon_G(X)$ is $G$-pseudocompact, because $X$ is) and because $X\subseteq\upsilon_GX\subseteq\beta X$ we have $\beta(\upsilon_GX)=\beta X=\beta_GX$. No apply the last sentence of my answer. | |
| Nov 28 at 12:56 | comment | added | Mehmet Onat | Do you think that my definition may not require $X\subset \upsilon _{G}X\subset \beta _{G}X$ ?. My second question: if $\upsilon _{G}X$ is $G$-pseudocompact and $\upsilon _{G}X$ is $G$-Hewitt, why $\upsilon _{G}X=\beta _{G}X$? | |
| Nov 28 at 11:51 | comment | added | KP Hart | @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 $\upsilon_GX=\beta_GX$. | |
| Nov 28 at 11:22 | comment | added | Mehmet Onat | 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 $X=\upsilon _{G}X$? | |
| Nov 28 at 11:21 | comment | added | Mehmet Onat | 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 $. | |
| Nov 28 at 11:20 | comment | added | Mehmet Onat | 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. | |
| Nov 28 at 11:19 | comment | added | Mehmet Onat | First of all, thank you very much for your reply. I would like to ask another question about this. | |
| Nov 28 at 10:17 | history | answered | KP Hart | CC BY-SA 4.0 |