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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
For a nonempty subset of with , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
Statement
Let and let be nonempty. The following are equivalent.
- is compact.
- is closed and bounded.
- is pseudocompact.
- Every continuous attains a maximum and a minimum on .
This theorem is a ZF statement. The nonemptiness hypothesis is necessary for condition 4, because the empty image has neither a maximum nor a minimum.
Facts & Assumptions
Given: A nonempty subset with , carrying the Euclidean subspace topology.
In Euclidean space, a subset is compact if and only if it is closed and bounded (Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).
A pseudocompact Euclidean subset is bounded and is closed (A pseudocompact subset of is bounded, A pseudocompact subset of is closed).
A continuous real-valued map on a nonempty compact topological space attains a maximum and a minimum (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, claim 2).
Pseudocompactness means that every continuous real-valued function has bounded image (Pseudocompact space: every continuous real-valued function has bounded image).
Compactness for the Euclidean metric and for its metric topology is the same condition (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).
Proof
Conditions 1 and 2 are equivalent by [L1] and [L5].
Condition 3 implies condition 2 by [L2].
Suppose condition 1 holds. Every continuous then attains a maximum and a minimum by [L3], so condition 4 holds.
Suppose condition 4 holds. For every continuous , its maximum and minimum bound , so is pseudocompact and condition 3 holds.
The implications , , and prove all four conditions equivalent.
Depends on
- A pseudocompact subset of $\mathbb{R}^n$ is bounded
- A pseudocompact subset of $\mathbb{R}^n$ is closed
- Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- 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
- Pseudocompact space: every continuous real-valued function has bounded image
- 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
Used by
- Assuming AC_ω and DC, compactness, sequential compactness, countable compactness, limit point compactness, completeness and total boundedness, pseudocompactness, closedness and boundedness, and the extreme-value property are equivalent for nonempty subsets of ℝⁿ with n≥1 Corollary
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact Corollary
- The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary Corollary
- ℝⁿ is closed and unbounded and is not compact for n≥1 Counterexample
- The open unit ball in ℝⁿ is bounded and not compact Counterexample
- Borel change of variables from the compact-support formula and Radon uniqueness Lemma
- Strict linear alternative for GCM trichotomy Lemma
Dependency tree · two levels
53 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
- Heine-Borel theorem (standard reference, not scraped)
- Extreme value theorem (standard reference, not scraped)