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If $A$ is a discrete, closed and $C$-embedded subset of a completely regular Hausdorff space $X$. Then how can we prove that for every continuous function $f:A\rightarrow (0,\infty)$, there exists a continuous extension $F:X\rightarrow (0,\infty)$ of $f$.

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  • $\begingroup$ What does "$C$-embedded" mean? My google-fu failed me. $\endgroup$ – Daniel Fischer Jan 23 '14 at 19:02
  • $\begingroup$ A subset $A$ of a topological space $X$ is called C-embedded in $X$ if every real-valued continuous function on $A$ can be extended to a continuous function on $X$. $\endgroup$ – shane Jan 24 '14 at 17:50
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A subset $A$ of a topological space $X$ is called C-embedded in $X$ if every real-valued continuous function on $A$ can be extended to a continuous function on $X$.

So we know that we can extend any continuous $f \colon A \to (0,\infty)$ to a continuous $F\colon X \to \mathbb{R}$, and the only problem is that the extension might attain values $\leqslant 0$.

If $f$ is strictly positive on $A$, then $g\colon A\to \mathbb{R}$ given by

$$g(a) = \log f(a)$$

is a continuous real valued function on $A$, and hence can be extended to a continuous real-valued $G \colon X \to \mathbb{R}$. Then $F(x) = e^{G(x)}$ is the desired strictly positive continuous extension of $f$.

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