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The Samuel reflection of a nonempty indiscrete uniform space is a singleton

Example

Let $X\ne\varnothing$ carry the indiscrete uniformity $\{X\times X\}$ . Its Samuel completion, equivalently its Hausdorff Samuel reflection, is a singleton. This is not called a compactification unless $X$ itself is a singleton.

Facts & Assumptions

Given: A nonempty set $X$ with only $X\times X$ as an entourage.

[L1]

A Samuel coordinate is a uniformly continuous function $f:X\to[0,1]$ (The Samuel uniformity generated by bounded uniformly continuous functions).

[L2]

A Hausdorff completion has separated target and dense uniformly continuous canonical map. Its induced topology is Hausdorff, hence $T_1$ , so its singletons are closed (A Hausdorff completion of a uniform space and its canonical dense map, Separated uniformity: the intersection of all entourages is the diagonal, A uniformity is separated if and only if its induced topology is Hausdorff, The implications proved on this page: perfectly normal gives completely normal under countable choice, and completely normal gives normal; normal with $T_1$ gives $T_3$ ; completely regular gives regular; regular with $T_1$ gives Urysohn, hence Hausdorff, hence $T_1$ , hence $T_0$ ; and metrizable gives every one of them, A space is $T_1$ if and only if every singleton is closed, if and only if every finite subset is closed, if and only if its topology contains the cofinite topology, The Samuel completion and, when compactifying, the Samuel compactification).

Verification

technique · direct

1.1

If $f:X\to[0,1]$ is uniformly continuous and $x,y\in X$ , then for every $\varepsilon>0$ the sole source entourage forces $|f(x)-f(y)|<\varepsilon$ ; hence $f(x)=f(y)$ .

2.1

Every Samuel coordinate is constant by step 1.1, so every Samuel pseudometric vanishes and the Samuel uniformity is again indiscrete.

L1step 1.1

3.1

Let $\eta:X\to S(X)$ be a Hausdorff completion of the Samuel uniformity. If $\eta(x)\ne\eta(y)$ , separatedness in [L2] gives a target entourage excluding that pair, while uniform continuity pulls it back to the sole source entourage $X\times X$ , a contradiction. Thus $\eta[X]$ is a singleton.

L2step 2.1

4.1

The nonempty singleton $\eta[X]$ is dense by [L2] and closed by [L2], so it is all of $S(X)$ . Thus the Samuel reflection is a singleton; if $X$ has at least two points its canonical map is not injective.

L2step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 117 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Megrelishvili, Samuel and Smirnov compactifications (standard reference, not scraped)