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I found in some proofs the following argument, which I do not understand: "suppose $X$ is Thyconoff and $\eta:X\to \beta X$ is the canonical map into its Stone–Čech compactification, since $X$ is Thyconoff we can consider $\eta^{-1}:\beta X\to X$..." and then the argument continues using $\eta^{-1}$ as a continuous retraction of $\eta$.

Is this correct? It seems to me that for an infinite discrete space (and hence Thyconoff) the map $\eta$ can not have any continuous retraction, more generally there are no continuous surjections from a compact space to an infinite discrete space.

Am I wrong?

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    Indeed, the author is wrong. $\eta^{-1}$ is a well defined continous mapping only on the image $\eta(X)$. He's actually always wrong if $X$ is not compact (the other case being trivially true since $\beta X = X$). Obviously non-compact space cannot be a continous image of a compact space. – 2017-02-01
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    Thank you Freakish, now I am more relaxed :-) !!! – 2017-02-01

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