Compactness

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✓ 22 results · all verified · 11 also independently AI-judged

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Compactness

3 · Items

Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it

For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide

A space is compact exactly when every family of closed subsets with the finite intersection property has nonempty intersection

A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact

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 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

Tube lemma: if K is compact and an open N ⊆ X × Z contains K × {z₀}, then N contains K × W for some open W ∋ z₀

A product of finitely many compact spaces is compact in the product topology

Alexander's subbase lemma: if every cover by members of a fixed subbasis has a finite subcover then the space is compact; the proof is an application of Zorn's lemma

Tychonoff's theorem: an arbitrary product of compact spaces is compact in the product topology, assuming the Axiom of Choice

Countably compact, Lindel"of, sequentially compact, limit point compact and σ-compact spaces, and relatively compact subsets

A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology

Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space

Dependent choice along a sequence of relations: if Rₙ is entire on A for every n, then from any a there is a sequence with aₙ Rₙ aₙ₊₁

Baire space: a topological space in which every countable intersection of dense open subsets is dense

Assuming dependent choice, every locally compact Hausdorff space is a Baire space

The one-point (Alexandroff) compactification X^* = X ∪ {∞}, whose open sets are the open sets of X together with the complements in X^* of the closed compact subsets of X

X^* is compact and contains X as an open subspace; X is dense in X^* exactly when X is not compact; and X^* is Hausdorff exactly when X is locally compact and Hausdorff

On an ordinal with its order topology the sets [0,β] and (α,β] form a basis of clopen sets, the isolated points are exactly the non-limit ordinals, and the space is Hausdorff

Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, ω₁ is countably compact and sequentially compact while ω₁ + 1 is compact

Every closed initial segment of the long ray is compact; the long ray is not compact; and, assuming countable choice, it is countably compact and not Lindel"of

In a compact Hausdorff space every quasicomponent is connected, so quasicomponents and components coincide

The quasicompact convention, why compactness of a subset is read intrinsically here, and what each result on this page costs in choice

FALSE: a compact subset of a topological space is closed

FALSE: every compact space is sequentially compact

FALSE: every sequentially compact space is compact

FALSE: every countably compact space is compact

FALSE: every subspace of a locally compact space is locally compact