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### DefinitionsUpDw('Definitions','-Abz-');

1. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology.
2. A uniform space is compact if and only if it is complete and totally bounded. (Web site)
3. A uniform space is metrizable if and only if has a countable base.
4. If a uniform space is metrizable, then the equalities and are also equivalent.
5. If a uniform space is an everywhere-dense subset of a uniform space, then.

### Linear Topological SpaceUpDw('LINEAR_TOPOLOGICAL_SPACE','-Abz-');

1. Every linear topological space (metrizable or not) is also a uniform space. (Web site)

### Uniform ContinuityUpDw('UNIFORM_CONTINUITY','-Abz-');

1. Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space. (Web site)
2. The δ-ε definition of uniform continuity translates directly into the uniform space definition.

### VersionUpDw('VERSION','-Abz-');

1. This is version 8 of uniform space, born on 2002-06-10, modified 2007-02-19.

### WayUpDw('WAY','-Abz-');

1. In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences. (Web site)

### HenceUpDw('HENCE','-Abz-');

1. The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space. (Web site)

### ConverselyUpDw('CONVERSELY','-Abz-');

1. Conversely, a uniform space is called complete if every Cauchy filter converges. (Web site)

### General TopologyUpDw('GENERAL_TOPOLOGY','-Abz-');

1. He introduced the concept of uniform space in general topology. (Web site)

### Metric SpacesUpDw('METRIC_SPACES','-Abz-');

1. One can also construct a completion for an arbitrary uniform space similar to the completion of metric spaces. (Web site)

### UniformlyUpDw('UNIFORMLY','-Abz-');

1. A uniform space X is uniformly connected if every uniformly continuous function from X to a discrete uniform space is constant.

### BoundedUpDw('BOUNDED','-Abz-');

1. General topology A uniform space is compact if and only if it is complete and totally bounded.
2. Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. (Web site)

### PlanetmathUpDw('PLANETMATH','-Abz-');

1. PlanetMath: separated uniform space (more info) Math for the people, by the people.

### Topological SpaceUpDw('TOPOLOGICAL_SPACE','-Abz-');

1. From the formal point of view, the notion of a uniform space is a sibling of the notion of a topological space.

### Uniform StructureUpDw('UNIFORM_STRUCTURE','-Abz-');

1. In the mathematical field of topology, a uniform space is a set with a uniform structure. (Web site)
2. If X is Tychonoff, then the uniform structure can be chosen so that β X becomes the completion of the uniform space X. (Web site)

### CompletenessUpDw('COMPLETENESS','-Abz-');

1. Remark. This is a generalization of the concept of completeness in a metric space, as a metric space is a uniform space.

### Metric SpaceUpDw('METRIC_SPACE','-Abz-');

1. Metric spaces A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
2. A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform space s. (Web site)
3. Generalizations of metric spaces Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. (Web site)

### Cauchy SequenceUpDw('CAUCHY_SEQUENCE','-Abz-');

1. A metric space (or uniform space) is complete if every Cauchy sequence in it converges. (Web site)
2. A Cauchy sequence in a uniform space is a sequence in whose section filter is a Cauchy filter.

### Topological Vector SpaceUpDw('TOPOLOGICAL_VECTOR_SPACE','-Abz-');

1. As with any topological vector space, a locally convex space is also a uniform space. (Web site)

### CompleteUpDw('COMPLETE','-Abz-');

1. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology. (Web site)
2. PlanetMath: complete uniform space (more info) Math for the people, by the people.
3. This is version 4 of complete uniform space, born on 2007-02-12, modified 2007-03-10.

### Complete Metric SpaceUpDw('COMPLETE_METRIC_SPACE','-Abz-');

1. Analogous to the notion of complete metric space, one can also consider completeness in a uniform space.

### Uniform ConvergenceUpDw('UNIFORM_CONVERGENCE','-Abz-');

1. The most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space. (Web site)

### EntourageUpDw('ENTOURAGE','-Abz-');

1. A uniform space X is discrete if and only if the diagonal {(x, x) : x is in X } is an entourage.
2. The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage. (Web site)

### Metrizable Uniform SpaceUpDw('METRIZABLE_UNIFORM_SPACE','-Abz-');

1. A metrizable uniform space is complete if and only if the metric generating its uniformity is complete.

### MetrizableUpDw('METRIZABLE','-Abz-');

1. Every uniform space of weight can be imbedded in a product of copies of a metrizable uniform space.
2. The topology of a metrizable uniform space is paracompact, by Stone's theorem.
3. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.

### Topological GroupsUpDw('TOPOLOGICAL_GROUPS','-Abz-');

1. Generalising both the metric spaces and the topological groups, every uniform space is completely regular. (Web site)

### Topological GroupUpDw('TOPOLOGICAL_GROUP','-Abz-');

1. Completion Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
2. Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.

### SpaceUpDw('SPACE','-Abz-');

1. Conversely, any completely regular space X can be made into a uniform space in some way. (Web site)
2. A metric space (or more generally any first-countable uniform space) is compact if and only if every sequence in the space has a convergent subsequence. (Web site)
3. In other words, every uniform space has a completely regular topology and every completely regular space X is uniformizable. (Web site)

### UniformityUpDw('UNIFORMITY','-Abz-');

1. One of the generalizations of a uniformity — the so-called -uniformity — is connected with the presence of the topology on a uniform space.
2. The weight of a uniform space is the least cardinality of a base of the uniformity.
3. A separable uniform space whose uniformity has a countable basis is second-countable.

### Uniform SpaceUpDw('UNIFORM_SPACE','-Abz-');

1. Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. (Web site)
2. In particular, topological vector spaces are uniform space s and one can thus talk about completeness, uniform convergence and uniform continuity.
3. The fine uniformity is characterized by the universal property: any continuous function f from a fine space X to a uniform space Y is uniformly continuous.

### CategoriesUpDw('Categories','-Abz-');

1. Science > Mathematics > Topology > Metric Space
2. Uniformity
3. Topological Group
4. Metrizable
5. Entourage
6. Books about "Uniform Space" in Amazon.com  Short phrases about "Uniform Space"   Originally created: April 08, 2008.   Links checked: April 21, 2013.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
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