Alphabeta Math

Compactness: Examples and Counterexamples

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✓ 8 results · all verified · 3 also independently AI-judged
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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies placed in the compactness hierarchy

Example

[0,1] and the Cantor set are compact, by Heine-Borel and by closedness inside [0,1]; and, assuming the Axiom of Choice, so is [0,1]^ℕ, by Tychonoff

Example

ℝ^* is homeomorphic to the unit circle by inverse stereographic projection, and ℕ^* is the ordinal space ω + 1

Example

ℝ and ℚ are σ-compact, and Lindelöf assuming countable choice; ℝ is locally compact and ℚ is nowhere locally compact

Example

ℝ with the half-open intervals [a,b) as a basis is not compact and, assuming the Axiom of Countable Choice, is Lindel"of, while its square is not Lindel"of, the antidiagonal being an uncountable closed discrete subspace

Counterexample

ℕ × {a,b} with the indiscrete topology on the second factor is limit point compact and not countably compact, so the hypothesis that singletons are closed is not decoration

Counterexample

Collapsing the set of naturals inside ℝ to a point gives a quotient of ℝ that is not locally compact at the collapsed point

Example

Assuming the Axiom of Choice, compactness of [0,1] derived from the subbase lemma alone, using only the rays as a subbasis and the least upper bound property

Sources