# A question about $C^{*}$-embedded and $C$-embedded

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.

• 1 implies 2 is Urysohn's lemma. Dec 26, 2018 at 23:12

## 1 Answer

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.