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The Stone–Čech compactification of a compact Hausdorff space adds no points
Statement
If is a Stone–Čech compactification of a compact Hausdorff space , then . Thus identifies homeomorphically with .
Facts & Assumptions
Given: A compact Hausdorff space and a Stone–Čech compactification of .
The continuous image of a compact space is compact, and a compact subset of a Hausdorff space is closed (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones).
Proof
The image is compact by [L1], hence closed in the Hausdorff space by [L1].
By the compactification condition in The Stone–Čech compactification by its compact-Hausdorff extension property, is dense in . A closed dense subset equals , so .
Depends on
- The Stone–Čech compactification by its compact-Hausdorff extension property
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$; the closure is the smallest closed superset and equals $A$ together with its derived set
Used by
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Sources
- E. Moorhouse, The Stone–Čech Compactification (standard reference, not scraped)