The classical theorem states that an Hausdorff space admits a Stone-Cech compactification if and only if it is completely regular or Tychnoff (i.e. such that any point $x$ and closed set $C$ with $x\not\in C$ can be separated by a continuous function with values in $[0;1]$ with $f(x)=0$ and $f\restriction C=1$). The following proof
gives the construction of the Stone-Cech compactification of any topological space. The proof shows that $X$ can be mapped (injectively if $X$ is Hausdorff) in a compact space with the universal property given by its Stone-Cech compactification (if $X$ has one). Otherwise the map most likely is not a topological embedding: it is certainly not injective if $X$ is not Hausdorff, and most likely it should not be an homeomorphism with the image if $X$ is not Tychonoff. I do not see in the proof how the assumption that X is Tychonoff is necessary and sufficient to grant that the embedding of $X$ into its Stone-Cech compactification is a homeomorphism with the image; nor I see any gap in the proof. Is there a simple example of a non-Tychonoff Hausdorff space which does not admit a Stone-Cech compactification? Thanks for the help.