Compactification of a topological semigroup with right-identity structure

Asked Nov 01 '22 at 16:20

Active Nov 03 '22 at 09:38

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I'm looking for a hint on this question:

Let $T$ be as a semigroup with right-identity structure i.e. $rs=r$ for all $r, s\in T$. Also, we consider $T$ as a topological semigroup that is a locally compact, noncompact Hausdorff space. Define $X=T\cup \{\infty \}$ to be the one-point compactification of $T$.

Also, assume that $U$ be an open set of $\infty$. Is there a compact set $K$ in $T$ such that $(U-\{\infty\})\subseteq K$?

Any help is welcome. Thanks in advance.

edited Nov 03 '22 at 09:38

asked Nov 01 '22 at 16:20

user479859

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What is the role of $e$? Is $U$ an open set of $X$? – J.-E. Pin Nov 03 '22 at 03:12
@J.-E.Pin, $e\in T$ is means that $T$ is monoid. Yes $U$ is an open set of $X$. – user479859 Nov 03 '22 at 03:41
Please edit your question to clarify. Moreover, if $e$ is the identity, $es = s$ for all $s$, contrary to your assumption. – J.-E. Pin Nov 03 '22 at 08:28
@J.-E.Pin, Thanks. in my question, I assume that $T$ is a topological semigroup and I edited it. – user479859 Nov 03 '22 at 09:40

Compactification of a topological semigroup with right-identity structure

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