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

$C^{*}(X)= \{ f \in C(X) | f \quad is \quad bounded \}$

Theorem 1: A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if only if any two completely separated sets in $S$ are completely separated sets in $X$.

Theorem 2: A $C^{*}$-embedded is $C$-embedded if only if it is completely separated from every zero-set disjoint from it.

According to the two above-mentioned theorems can be shown below problem? can you help me?

The following are equivalent for any Hausdorff space $X$.

1: $X$ is normal.

2:Any two disjoint closed sets are completely separated.

3:Every closed set is $C^{*}$-embedded.

4:Every closed set is $C$-embedded.

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    $\begingroup$ 1 implies 2 is Urysohn's lemma. $\endgroup$ Dec 26, 2018 at 23:12

1 Answer 1

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Conditions 1-4 are equivalent. Implications $4\Rightarrow 3$, $3\Rightarrow 2$, and $2\Rightarrow 1$ are obvious. Implication $1\Rightarrow 2$ follows from Urysohn’s lemma [Eng, 1.5.11 and p.42]. $1\Rightarrow 3,4 $ by the Tietze-Urysohn Theorem [Eng, 2.1.8]. $2\Rightarrow 3$ also follows from Theorem 1, and $3\Rightarrow 4$ from Theorem 2.

References

[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

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