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I am wondering how to prove the following theorem:

Let "X" be a Tychonoff space. Then its Stone-Čech compactification exists and it is unique (up to homeomorphism).

The uniqueness part is clear to me, but what seems difficult is check that $\beta X$ is really a compactification and that it satisfies the universal property.

I am not asking for whole proof here, but it would be great if you provided a source where I can study this proof.

I have found a proof in a material by Ivan Khatchatourian (Theorem 5.4) , but he refers to various sources used in the proof which are not attached to the material, so it is not that helpful.

Tereza Tizkova
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    A thorough presentation in the chapter Compact Spaces in the book General Topology by R. Engelking. – DanielWainfleet Jun 26 '22 at 11:27
  • I would recommend the exposition in Rings of continuous functions by Gillman and Jerison, chapter 6. They construct $\beta X$ as the set of $z$-ultrafilters on $X$. Later in the book other constructions are also present e.g. by embedding $X$ into $\prod_{f\in C_b(X)}\mathbb{R}$ or using spectrum of rings $C(X)$ or $C_b(X)$. – Jakobian Dec 08 '24 at 12:30

1 Answers1

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I am trying to answer my own question, so feel free to correct me. As you can see, the final part (showing the existence) is still missing, because I have not found or thought of a sufficient proof that I understand.

Uniqueness

For uniqueness, we first prove that if a compactification of a Tychonoff space $X$ satisfying the given universal property exists, then it is unique up to equivalence. Suppose we have two compactifications of $X$ satisfying the property, namely $\gamma_1 : X \rightarrow \gamma_1 X$ and $\gamma_2 : X \rightarrow \gamma_2 X$.

Then $\gamma_1$ and $\gamma_2$ have to be continuous functions from $X$ into compact Hausdorff spaces, and so by the universal property we can find other continuous functions $f: \gamma_1 X \rightarrow \gamma_2 X$ and $g: \gamma_2 X \rightarrow \gamma_1 X$ such that $\gamma_2 = f \circ \gamma_1$ and $\gamma_1 = g \circ \gamma_2$. From this it immediately follows that $g \circ f$ is the identity map on $\gamma_1 X$, and that $f \circ g$ is the identity map on $\gamma_2 X$.

Therefore $f$ is a continuous function with a continuous inverse, making it the homeomorphism we require.

Existence & universal property

Now we show that for any continuous mapping $f: X \rightarrow K$ where $K$ is a compact space, we can find an extension to a continuous mapping $\beta f: \beta X \rightarrow K$, that is, we show the universal property of the Stone-Čech compactification. Denote by $\gamma$ the embedding of $X$ in its arbitrary compactification $\gamma X$ and identify $X$ with the subspace $\gamma(X)$ of $\gamma X$.

We can use the diagonal theorem: For a family $\{f_s\}_{s \in S}$ of continuous mappings, where $f_s: X \rightarrow Y_s$ separates points for each $s \in S$, then the diagonal $\Delta_{s \in S}f_s: X \rightarrow \prod_{s \in S} Y_s$ is a one-to-one mapping. Moreover, if the said family separates points and closed sets, then $\Delta_{s \in S}f_s$ is a homeomorphic embedding. This theorem implies that $\beta \Delta f: X \rightarrow \beta X \times K$ is a homeomorphic embedding. Hence, $\overline{\gamma(X)} \subset \beta X \times K$ is a compactification of $X$ and by the maximality of $\beta X$, we can find a continuous mapping $g: \beta X \rightarrow \gamma X$ such that $g \beta = \gamma$.

Now, let $p: \gamma X \rightarrow K$ be the restriction of the projection of $\beta X \times K$ onto $K$ to $\gamma X$. Then the mapping $p \circ g: \beta X \rightarrow K$ is the desired extension of $f$, since $p \circ g \beta = p \circ \gamma = f$. \par Complete proof of the existence of the Stone-Čech compactifciation can be found for example in Engelking´s General Topology (as was suggested in the comments by @DanielWainfleet).

Tereza Tizkova
  • 2,750