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we shall say that a subspace $S$ of $X$ is $C$-embedded ($C^{*}$-embedded) in $X$ if every function in $C(S)$ ($ C^{*}(S) $) can be extended to a function in $C(X)$($C^{*}(X)$).

Urysohn extension theorem: A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if only if any two completely separated sets in $S$ are completely separated in $X$.

a space $X$ is said to be extremely disconnected if every open set has an open closure. $X$ is basically disconnected if every cozero-set has an open closure. hence any extremely disconnected space is basically disconnected

we know that$X$ is extremely disconnected if and only if every pair of disjoint open sets have disjoint closures.

so, my questions are:

1: $X$ is extremely disconnected if and only if every open subspace is $C^{*}$ - embedded.

2:Every dense subspace $X$ of an extremely disconnected space $T $ is extremely disconnected. In fact, disjoint open not sets in $X$ have disjoint open closure in $T$.

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1 Answer 1

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Fact 2. follows from the general fact:

Let $X$ be a space, $U$ open in $X$, $D$ dense in $X$. Then $\overline{D \cap U} = \overline{U}$.

$D \cap U$ is the general form of an open subset of $D$. So by $$\operatorname{cl}_D(U \cap D) = \operatorname{cl}_X(U \cap D) \cap D = \operatorname{cl}_X(U) \cap D$$

its closure is clopen in $D$.

Fact 1. Let $X$ be extremally disconnected and $O$ be an open subspace of $X$. Let $A$ and $B$ be completely separated subsets of $O$. Then they are separated by two disjoint open sets (of $O$, and hence of $X$). In an extremally disconnected space disjoint open sets are completely separated (fact 2 in your question here). Apply this to $X$ and we have that $A$ and $B$ are completely separated in $X$. Urysohn then implies that $O$ is $C$-embedded in $X$. The reverse is trivial: if $U$ and $V$ are disjoint open then $O=U \cup V$ is $C$-embedded and $U$ and $V$ are completely separated in $O$ and hence by $C$-embeddedness have disjoint closures in $X$.

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