Alphabeta MathUniform Spaces: the Three Definitions
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✓ 38 results · all verified · 21 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Uniform space in the entourage formulation
Lemma
Every uniformity has a base of symmetric entourages
Theorem
The sets containing an entourage ball about each of their points form a topology
Definition
Separated uniformity: the intersection of all entourages is the diagonal
Theorem
A uniformity is separated if and only if its induced topology is Hausdorff
Definition
Uniformly continuous map between uniform spaces
Theorem
Every uniformly continuous map is continuous for the induced topologies
Lemma
A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated
Definition
Uniform space in the uniform-cover formulation
Lemma
On a nonempty set, entourage uniformities and uniform-cover structures determine one another
Definition
A gauge of pseudometrics and, on a nonempty set, the uniformity it generates
Lemma
Assuming dependent choice, every entourage admits a normal symmetric sequence subordinate to it
Lemma
A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls
Theorem
Assuming dependent choice, every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics
Theorem
On a nonempty set, entourages and uniform covers give equivalent definitions of a uniform space in ZF, and under dependent choice they are also equivalent to gauges of pseudometrics
Definition
Cauchy filter in a uniform space
Lemma
Every convergent filter on a uniform space is Cauchy
Lemma
A Cauchy filter with a cluster point converges to that point
Definition
Complete uniform space: every Cauchy filter converges
Definition
Uniform embedding and uniform isomorphism
Definition
A Hausdorff completion of a uniform space and its canonical dense map
Lemma
Every Cauchy filter canonically determines a unique minimal Cauchy filter coarser than it
Lemma
The standard entourages on minimal Cauchy filters form a separated uniformity
Lemma
The uniform space of minimal Cauchy filters is complete
Lemma
The minimal Cauchy filters associated to points define a uniformly continuous dense canonical map
Theorem
Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated
Theorem
Every uniformly continuous map into a complete Hausdorff uniform space extends uniquely across the Hausdorff completion; consequently completions are unique up to a unique uniform isomorphism
Definition
Totally bounded uniform space
Lemma
Every ultrafilter on a totally bounded uniform space is Cauchy
Lemma
Every compact uniform space is complete
Lemma
Every compact uniform space is totally bounded
Theorem
Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact
Corollary
Assuming the ultrafilter lemma, a uniform space is compact if and only if it is complete and totally bounded
Lemma
Every open cover of a compact Hausdorff space has a finite open star-refinement
Theorem
The covers admitting an open refinement form a compatible uniform-cover structure on a nonempty compact Hausdorff space; in particular every open cover is uniform
Theorem
A nonempty compact Hausdorff space carries exactly one compatible uniformity
Corollary
Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous
Definition
A uniformity with a countable entourage base
Lemma
A countable entourage base can be replaced in ZF by a decreasing symmetric base whose next triple composite lies in the preceding member
Theorem
Every countably based uniformity is generated by one pseudometric, which is a metric exactly when the uniformity is separated
Definition
Uniformizable and separated-uniformizable topological spaces
Lemma
Every uniformizable space is regular
Lemma
Assuming dependent choice, every uniformizable space is completely regular
Lemma
The topology of a nonempty completely regular space is induced by the gauge of its continuous [0,1]-valued pseudometrics
Theorem
Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular
Corollary
Assuming dependent choice, a nonempty topological space is separated-uniformizable if and only if it is Tychonoff
Definition
Topological group: multiplication and inversion are continuous
Lemma
Left and right translations and inversion in a topological group are homeomorphisms
Definition
The left and right uniformities of a topological group
Theorem
The left and right uniformities of a topological group induce its topology, and inversion interchanges them
Definition
The upper and Roelcke uniformities generated from the left and right uniformities of a topological group
Corollary
Every topological group is uniformizable, and assuming dependent choice it is completely regular
Definition
The pointwise and uniform-convergence uniformities on a function set Y^X
Lemma
The uniform-convergence uniformity is finer than the pointwise uniformity, and they agree when the domain is finite
Sources
- J. Wodzicki, Uniform Structure
- Encyclopedia of Mathematics, Uniform space
- M. Megrelishvili, Lecture Notes in Topological Groups
- M. Kunzinger, General Topology
- Encyclopedia of Mathematics, Complete uniform space
- J. H. V. Hunt, Boletín de la Sociedad Matemática Mexicana 34 (1989), 11–21
- C. Rosendal, Coarse Geometry of Topological Groups