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The Stone–Čech compactification by its compact-Hausdorff extension property
Definition
A Stone–Čech compactification of is a Hausdorff compactification (A Hausdorff compactification as a dense embedding into a compact Hausdorff space) such that for every compact Hausdorff space and continuous map (Continuity of a map of topological spaces at a point and globally), there is a unique continuous with . The universal property, rather than a particular construction, is the definition.
Depends on
Used by
- Stone–Čech compactifications are uniquely homeomorphic over the original space Corollary
- The Stone–Čech compactification of a compact Hausdorff space adds no points Corollary
- Under dependent choice and the ultrafilter lemma, the Stone-Cech compactification maps continuously onto the Samuel compactification Corollary
- The one-point compactification of discrete ℕ is not βℕ Counterexample
- Every function from discrete ℕ to [0,1] extends uniquely to βℕ Example
- Stone duality for a power set algebra Example
- Under dependent choice and the ultrafilter lemma, the Samuel compactification of the discrete natural numbers is beta N Example
- FALSE: every Hausdorff compactification has the Stone–Čech extension property False statement
- 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 dependent choice, Stone-Cech compactification is left adjoint to the compact-Hausdorff inclusion Theorem
- Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification Theorem
Dependency tree · two levels
9 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
- Stacks Project, Stone–Čech compactification (standard reference, not scraped)