TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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For a nonempty subset of $\mathbb{R}^n$ with $n\ge1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent

Statement

Let $n\ge1$ and let $A\subseteq\mathbb{R}^n$ be nonempty. The following are equivalent.

$A$ is compact.
$A$ is closed and bounded.
$A$ is pseudocompact.
Every continuous $f:A\to\mathbb{R}$ attains a maximum and a minimum on $A$ .

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 $A\subseteq\mathbb{R}^n$ with $n\ge1$ , carrying the Euclidean subspace topology.

[L1]

In Euclidean space, a subset is compact if and only if it is closed and bounded (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).

[L2]

A pseudocompact Euclidean subset is bounded and is closed (A pseudocompact subset of $\mathbb{R}^n$ is bounded, A pseudocompact subset of $\mathbb{R}^n$ is closed).

[L3]

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).

[L4]

Pseudocompactness means that every continuous real-valued function has bounded image (Pseudocompact space: every continuous real-valued function has bounded image).

[L5]

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

technique · direct

1.1

Conditions 1 and 2 are equivalent by [L1] and [L5].

L1L5

1.2

Condition 3 implies condition 2 by [L2].

1.3

Suppose condition 1 holds. Every continuous $f:A\to\mathbb{R}$ then attains a maximum and a minimum by [L3], so condition 4 holds.

1.4

Suppose condition 4 holds. For every continuous $f:A\to\mathbb{R}$ , its maximum and minimum bound $f[A]$ , so $A$ is pseudocompact and condition 3 holds.

2.1

The implications $1\Leftrightarrow2$ , $3\Rightarrow2\Rightarrow1$ , and $1\Rightarrow4\Rightarrow3$ prove all four conditions equivalent.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 137 results over 19 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

Heine-Borel theorem (standard reference, not scraped)
Extreme value theorem (standard reference, not scraped)

For a nonempty subset of Rn\mathbb{R}^nRn with n≥1n\ge1n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent