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I am now working on Munkres Topology,Stone–Čech compactification part. He says that the correspondence between a completely regular space and its Stone–Čech compactification is a funtor. To verify this, I need to show that the correspondence preserves the identity mapping and composites of functions. It was easy to show the former, but I am not sure how to do the latter... The situation is:

Let $\beta(X)$ denote a Stone–Čech compactification of a topological space $X$.

Let $X,Y,Z$ be completely regular spaces.

Let $f:X\rightarrow Y$ , $g:Y\rightarrow Z$ be continuous maps.

Let $\beta(f):\beta(X) \rightarrow \beta(Y)$ extend $\iota \circ f$, where $\iota:Y \rightarrow \beta(Y)$ is an inclusion mapping.

What I need to show is that

$$\beta(g\circ f)=\beta(g) \circ\beta(f)$$

How can I show this? It seems obvious if $x\in X$. But how can I show for the case $x\in \beta(X)-X$? Any help would be really appreciated. Thanks.

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The continuous map $\beta(f):\beta(X)\to \beta(Y)$ fits into the diagram $$\require{AMScd}\begin{CD} X @> f >>Y\\ @V \iota_X VV @V \iota_Y VV \\\beta(X) @> \beta(f) >> \beta(Y) \end{CD}$$ and is the unique such map.

So we have $$\require{AMScd}\begin{CD} X @> g\circ f >>Z\\ @V \iota_X VV @V \iota_Z VV \\\beta(X) @> \beta(g\circ f) >> \beta(Z) \end{CD}$$

but also

$$\require{AMScd}\begin{CD} X @> f >>Y @> g >> Z\\ @V \iota_X VV @V \iota_Y VV @V \iota_Z VV\\\beta(X) @> \beta(f) >> \beta(Y) @> \beta(g) >> \beta(Z) \end{CD}$$ commutes as both small squares commute.

So both $\beta(g\circ f)$ and $\beta(g)\circ \beta(f)$ extend $\iota_Z \circ g\circ f$ so they must be equal by the uniqueness.

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    $\begingroup$ The unicity of $\beta(f)$ is easy to see as $\iota_X[X]$ is dense in $\beta X$ and the range $\beta Y$ is Hausdorff. On the dense $X$ $\beta(f)$ must equal $\iota_Y \circ f$ etc. I'm not sure whether Munkres covers this explicitly. $\endgroup$ – Henno Brandsma Feb 24 '17 at 5:21

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