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Under dependent choice and the ultrafilter lemma, the Samuel compactification of the discrete natural numbers is beta N

Example

Assume dependent choice and the ultrafilter lemma. Give $\mathbb N$ the metric $d(m,n)=0$ for $m=n$ and $d(m,n)=1$ otherwise. Its Samuel compactification is isomorphic over $\mathbb N$ to its Stone--Cech compactification $\beta\mathbb N$ .

Facts & Assumptions

Given: Dependent choice, the ultrafilter lemma, the set $\mathbb N$ , and the displayed zero-one function $d$ .

[L1]

A metric must satisfy separation, symmetry, and the triangle inequality; its metric entourages are $E_\varepsilon=\{(x,y):d(x,y)<\varepsilon\}$ , induce the metric topology, and form a separated uniformity (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated).

[L2]

Samuel compactifications extend precisely the uniformly continuous maps to compact Hausdorff targets, and Stone--Cech compactifications extend precisely the continuous maps to those targets (Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification, The Stone–Čech compactification by its compact-Hausdorff extension property).

[L3]

Under dependent choice, the separated metric uniformity makes the discrete topology on $\mathbb N$ Tychonoff; under the ultrafilter lemma its evaluation closure is a Stone--Cech compactification (A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated, 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).

[L4]

Under dependent choice and the ultrafilter lemma, the Samuel completion of this separated metric uniformity is a Samuel compactification (Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space).

Verification

technique · direct

1.1

The displayed $d$ satisfies the metric axioms: if $m\ne n$ , then for every $r$ at least one of $m\ne r$ or $r\ne n$ holds, which proves the triangle inequality; symmetry and separation are immediate.

2.1

The entourage $E_{1/2}$ is the diagonal. Therefore every map from $\mathbb N$ to a uniform space is uniformly continuous, since every target entourage contains the diagonal; every map from the discrete topology is continuous.

L1step 1.1

3.1

The two extension properties in [L2] consequently quantify over the same maps from $\mathbb N$ ; using the Samuel compactification whose existence is given by [L4], they yield inverse maps between $S(\mathbb N)$ and $\beta\mathbb N$ fixing $\mathbb N$ .

L2L4step 2.1

4.1

The Stone--Cech object exists by [L3] and the Samuel compactification by [L4]; the two inverse maps give the asserted isomorphism over $\mathbb N$ .

L3L4step 3.1∎

Depends on

Used by

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Sources

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