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Under dependent choice and the ultrafilter lemma, the Stone-Cech compactification maps continuously onto the Samuel compactification

Statement

Assume dependent choice and the ultrafilter lemma. If $X$ is a separated uniform space, then its evaluation-closure Stone--Cech compactification $j:X\to\beta X$ admits a continuous surjection

q:\beta X\longrightarrow S(X)

such that $qj=\eta$ , where $\eta:X\to S(X)$ is the Samuel compactification map.

Facts & Assumptions

Given: The stated choice principles and a separated uniform space $X$ .

[L1]

Under dependent choice, a separated uniformizable space is Tychonoff; under the two choice principles its evaluation closure is a Stone--Cech compactification (Assuming dependent choice, a nonempty topological space is separated-uniformizable if and only if it is Tychonoff, Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification).

[L2]

Under the same principles, the Samuel completion is a compactification, hence $S(X)$ is compact Hausdorff and $\eta[X]$ is dense (Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space).

[L3]

The Stone--Cech extension property extends a continuous map from $X$ to a compact Hausdorff target uniquely (The Stone–Čech compactification by its compact-Hausdorff extension property).

[L4]

A continuous image of a compact space is compact, and a compact subset of a Hausdorff space is closed (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones).

Proof

technique · direct

1.1

The map $\eta:X\to S(X)$ is continuous by [L2], so [L1] and [L3] give a continuous $q:\beta X\to S(X)$ with $qj=\eta$ .

L1L2L3

2.1

By [L1], $\beta X$ is compact. Thus the image $q[\beta X]$ is compact and therefore closed in the Hausdorff space $S(X)$ by [L4].

L1L4step 1.1

3.1

Since $q[\beta X]$ contains $qj[X]=\eta[X]$ , it contains a dense subset of $S(X)$ ; its closedness from step 2.1 gives $q[\beta X]=S(X)$ .

L2step 1.1step 2.1

4.1

Hence $q$ is the asserted continuous surjection.

step 3.1∎

Depends on

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Sources

Garrido and Meroño, The Samuel realcompactification (standard reference, not scraped)
Stacks Project, Stone-Cech compactification (standard reference, not scraped)