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A continuous map of compact Hausdorff spaces is an ultrafilter-algebra homomorphism

Statement

Let $f : X \to Y$ be continuous between compact Hausdorff spaces, and let $ξ_{X}, ξ_{Y}$ be their ultrafilter-limit maps. Then

$f ξ_{X} = ξ_{Y} β (f),$

so $f$ is an ultrafilter-algebra homomorphism.

Facts & Assumptions

Given: A continuous map $f : X \to Y$ of compact Hausdorff spaces and an ultrafilter $U$ on $X$ .

[L1]

The pushforward $f_{*} U$ is an ultrafilter on $Y$ , and ultrafilter pushforward is functorial (Pushforward sends ultrafilters to ultrafilters and is functorial).

[L2]

Every ultrafilter on a compact Hausdorff space has exactly one limit (A given ultrafilter on a compact Hausdorff space has a unique limit).

[L3]

A $T$ -algebra homomorphism $f : (A, a) \to (B, b)$ satisfies $f a = b T (f)$ (Algebra and algebra homomorphism for a monad).

Proof

technique · direct

1.1

Push $U$ forward along $f$ to the ultrafilter $f_{*} U$ on $Y$ supplied by [L1].

2.1

If $U$ converges to $x$ , then for each neighbourhood $V$ of $f (x)$ , continuity makes $f^{- 1} [V]$ a neighbourhood of $x$ and hence a member of $U$ . Therefore $V \in f_{*} U$ , so the pushforward converges to $f (x)$ .

step 1.1given

3.1

Taking $x = ξ_{X} (U)$ , uniqueness in the target gives $ξ_{Y} (f_{*} U) = f (ξ_{X} (U))$ .

step 2.1L2

4.1

Since $f_{*} U = β (f) (U)$ , step 3.1 is the equation $f ξ_{X} = ξ_{Y} β (f)$ in [L3]. Thus $f$ is an algebra homomorphism, including for the unique map from an empty compact space.

step 3.1L3∎

Depends on

Used by

Under the ultrafilter lemma, compact Hausdorff spaces and ultrafilter algebras are recovered by the two limit constructions Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Goubault-Larrecq, Algebras of filter-related monads: I. Ultrafilters and Manes' theorem (standard reference, not scraped)