Compactness in Metric Spaces

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

Open cover, subcover, compact metric space, and compact subset of a metric space

Lemma

A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it

Definition

Countably compact, sequentially compact and limit point compact metric spaces

Definition

Finite ε-net and totally bounded metric space

Lemma

A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded

Theorem

A metric space is compact if and only if every family of closed subsets with the finite intersection property has nonempty intersection

Lemma

In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle

Lemma

For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice

Theorem

Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ 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

Theorem

The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset

Theorem

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

Theorem

Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover

Theorem

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous

Theorem

A continuous bijection from a compact metric space onto a metric space carries open sets to open sets, so its inverse is continuous

Lemma

A compact metric space has a countable dense subset, by countable choice

Remark

What each implication between the compactness properties of a metric space costs: which are theorems of ZF, which use countable choice, and which use dependent choice

Example

With the discrete metric d(x,y) = 1 for x ≠ y, a space is compact iff it is totally bounded iff it is finite, and it is complete whatever its size

False statement

Compactness in Metric Spaces

3 · Items

Open cover, subcover, compact metric space, and compact subset of a metric space

Countably compact, sequentially compact and limit point compact metric spaces

Finite ε-net and totally bounded metric space

A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded

A metric space is compact if and only if every family of closed subsets with the finite intersection property has nonempty intersection

A closed subset of a compact metric space is compact

A compact subset of a metric space is closed and bounded

A compact metric space is complete and totally bounded, and neither implication uses any choice principle

In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle

A sequentially compact metric space is complete, with no choice principle used

A sequentially compact metric space is totally bounded, proved from the axiom of dependent choice

A complete, totally bounded metric space is compact, proved from countable choice used exactly once

For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice

Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ 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

The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous

A continuous bijection from a compact metric space onto a metric space carries open sets to open sets, so its inverse is continuous

A compact metric space has a countable dense subset, by countable choice

What each implication between the compactness properties of a metric space costs: which are theorems of ZF, which use countable choice, and which use dependent choice

With the discrete metric d(x,y) = 1 for x ≠ y, a space is compact iff it is totally bounded iff it is finite, and it is complete whatever its size

FALSE: a closed and bounded subset of a metric space is compact

FALSE: a totally bounded metric space is compact

FALSE: a bounded metric space is totally bounded