Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Pseudocompact space: every continuous real-valued function has bounded image
Definition
A topological space is pseudocompact when every continuous map (Continuity of a map of topological spaces at a point and globally) has bounded image: there are reals with for every (Lower bound, bounded below, bounded set).
A subset of a topological space is pseudocompact when , equipped with its subspace topology, is pseudocompact (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
Remarks
No separation axiom is part of this definition. Some texts reserve the word for completely regular spaces; here it names exactly the bounded-image condition just stated.
Depends on
Used by
- An infinite particular-point space is pseudocompact and not compact Counterexample
- A pseudocompact subset of ℝⁿ is bounded Lemma
- A pseudocompact subset of ℝⁿ is closed Lemma
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 16 results over 8 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
- Pseudocompact space (standard reference, not scraped)