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Čech-complete spaces as subspaces of Hausdorff compactifications
Definition
A Tychonoff space is Čech-complete when there is a Hausdorff compactification of (A Hausdorff compactification as a dense embedding into a compact Hausdorff space) for which is a subset of ( and subsets of a topological space, agreeing with the real-line notion). The definition asks for one compactification; under the ultrafilter lemma and Dependent Choice, Under the ultrafilter lemma and Dependent Choice, a Tychonoff space is in some Hausdorff compactification exactly when it is in every one proves the equivalent every-compactification form.
Depends on
Used by
- Every locally compact Hausdorff space is Čech-complete Corollary
- Closed subspaces of Čech-complete spaces are Čech-complete Proposition
- Under the Axiom of Choice, topological sums of Čech-complete spaces are Čech-complete Proposition
- Under Dependent Choice, every Čech-complete space is Baire Theorem
- Under the Axiom of Choice, countable products of Čech-complete spaces are Čech-complete Theorem
- Under the ultrafilter lemma and Dependent Choice, a Tychonoff space is G_δ in some Hausdorff compactification exactly when it is G_δ in every one Theorem
- Under the ultrafilter lemma and the Axiom of Choice, every completely metrizable space is Čech-complete Theorem
- Under the ultrafilter lemma, every metrizable Čech-complete space is completely metrizable Theorem
- Under the ultrafilter lemma, Frolík's internal open-cover characterisation of Čech-completeness Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 65 results over 14 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
- David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
- Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
- MFF General Topology course summary, §4.3 (standard reference, not scraped)
- Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)