Alphabeta Math

Uniform Spaces: the Three Definitions: Examples and Counterexamples

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✓ 8 results · all verified · 5 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The usual metric entourages on ℝ induce its usual topology and usual uniform continuity

Example

The map x↦ x/(1+|x|) is a uniformly continuous homeomorphism from ℝ to (-1,1) whose inverse is not uniformly continuous

False statement

FALSE: every uniformizable topology has a unique compatible uniformity

Example

The left, right, upper and Roelcke uniformities of the additive topological group ℝ all equal its metric uniformity

Example

The functions fₙ(k)=1 for k≥ n and 0 otherwise converge pointwise but not uniformly on ℕ

Example

The closed unit interval has exactly one compatible uniformity, namely its usual metric uniformity

Counterexample

The K-topology on ℝ is not uniformizable

Example

Assuming the Ultrafilter Lemma and Countable Choice, an uncountable Cantor cube is compact Hausdorff and uniformizable but not first countable, hence not metrizable

Sources