Alphabeta Math

Compactness in Metric Spaces: Examples and Counterexamples

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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The cube [-M,M]ⁿ in ℝⁿ is totally bounded, with an explicit finite ε-net of grid points and no appeal to the integer part

Example

In any metric space the range of a convergent sequence together with its limit is compact, worked out for {0} ∪ {1/(k+1) : k ∈ ℕ} in ℝ

Example

The distance from a point to a nonempty compact set is attained at a point of that set, and two disjoint compact sets are at positive distance

Example

The cover of [0,1] by (-1, 2/3) and (1/3, 2) has Lebesgue number 1/3, and no larger one

Counterexample

In the bounded real-valued functions on ℕ with the supremum metric, the closed unit ball is closed and bounded and is not compact: the indicator functions of the singletons are pairwise at distance 1

Counterexample

ℕ with the discrete metric is bounded and is not totally bounded

Counterexample

The open interval (0,1) is totally bounded and not compact, the cover by the intervals (1/(k+2), 1) having no finite subcover

Counterexample

On (0,1) the identity is bounded with no greatest value and x ↦ 1/x is continuous and unbounded, so the extreme value theorem needs compactness and not merely boundedness of the domain

Counterexample

x ↦ 1/x is continuous on (0,1) and not uniformly continuous, so Heine-Cantor needs compactness of the domain

Counterexample

The cover of (0,1) by the intervals (1/(k+2), 1) has no Lebesgue number, so the Lebesgue number lemma needs compactness

Sources