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Please help me to figure out this problems.

Let X be a metric space and let Y be a compact metric space. Denote by $\Phi: X \rightarrow \beta X$ the standard embedding of X as a dense subset of its Stone-Cech compactification $\beta X$.

With this assumptions, I want to prove that to every continuous map $f: X \rightarrow Y$ there corresponds a continuous map $f^\beta : \beta X \rightarrow Y$ so that $f^\beta \circ \Phi = f$.

I started the prove with f induces a map $H_f: C_b(X)^* \rightarrow C_b(Y)^*$; then I consider the restriction of $H_f$ to $\beta X$. Then I don't know what to do.

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  • Show first that if $f:X\to Y$ then there is an induced map $\beta f:\beta X\to\beta Y$ which is continuous.

  • Then show that if $Y$ is compact, the natural map $Y\to \beta Y$ is an homeomorphism.

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It is true when X is any Tychonoff space and Y is any compact Hausdorff space. For a proof, see General Topology (Revised and completed edition) by R.Engelking, section 3.6.1 (page 173 in the English edition).

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