﻿ "Space" related terms, short phrases and links

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This Review contains major "Space"- related terms, short phrases and links grouped together in the form of Encyclopedia article.

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

1. A space is a Baire space if the intersection of any countable collection of dense open sets is dense.
2. A space is a regular -space.
3. A space is a Baire space if any intersection of countably many dense open sets is dense.
4. A space is also known as a Hausdorff space.
5. The space is called the quotient space of the space with respect to .

### Whole SpaceUpDw('WHOLE_SPACE','-Abz-');

1. Equivalently, a set is dense if its closure is the whole space.
2. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open.

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

1. A space X is a Baire space if it is not meagre in itself.
2. Every hyper-connected space is connected.
3. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
4. A compact subset of a Hausdorff space is closed.

### Simply ConnectedUpDw('SIMPLY_CONNECTED','-Abz-');

1. A normal space is Hausdorff if and only if it is T 1. Normal Hausdorff spaces are always Tychonoff.
2. Every contractible space is simply connected.
3. Every simply connected space and every locally simply connected space is semilocally simply connected.
4. Relaxing the second requirement, or removing the third or fourth, leads to the concepts of a pseudometric space, a quasimetric space, or a semimetric space.
5. A topological space is connected, locally path connected, and semilocally simply connected if and only if it has a universal cover.

### Every OpenUpDw('EVERY_OPEN','-Abz-');

1. A space is almost discrete if every open set is closed (hence clopen).
2. On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.

### Disjoint NeighbourhoodsUpDw('DISJOINT_NEIGHBOURHOODS','-Abz-');

1. Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods.
2. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
3. As you might guess from the terminology, a fully normal space is normal.

### Local BaseUpDw('LOCAL_BASE','-Abz-');

1. A space is locally connected if every point has a local base consisting of connected sets.
2. A space is first-countable if every point has a countable local base.
3. A space is locally path-connected if every point has a local base consisting of path-connected sets.

### Completely NormalUpDw('COMPLETELY_NORMAL','-Abz-');

1. A space is completely normal if any two separated sets have disjoint neighbourhoods.
2. A perfectly normal space must also be completely normal.

### Every PointUpDw('EVERY_POINT','-Abz-');

1. A topological space is locally contractible is every point has a local base of contractible neighborhoods.
2. A space is locally metrizable if every point has a metrizable neighbourhood.
3. Every totally disconnected space is T 1, since every point is a connected component and therefore closed.

### Compact SpaceUpDw('COMPACT_SPACE','-Abz-');

1. Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
2. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.
3. A closed subset of a compact space is compact.

### Finite SubcoverUpDw('FINITE_SUBCOVER','-Abz-');

1. A space is countably compact if every countable open cover has a finite subcover.
2. The modern general definition calls a topological space compact if every open cover of it has a finite subcover.
3. If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact.

### Locally Path-ConnectedUpDw('LOCALLY_PATH-CONNECTED','-Abz-');

1. A locally path-connected space is connected if and only if it is path-connected.
2. Every metrizable space is first-countable.

### Lindel OumlUpDw('LINDEL_OUML','-Abz-');

1. A space is Lindelöf if every open cover has a countable subcover.
2. Every second-countable space is first-countable, separable, and Lindelöf.

### Weakly Countably CompactUpDw('WEAKLY_COUNTABLY_COMPACT','-Abz-');

1. Every countably compact space is pseudocompact and weakly countably compact.
2. A space is weakly countably compact (or limit point compact) if every infinite subset has a limit point.

### Regular HausdorffUpDw('REGULAR_HAUSDORFF','-Abz-');

1. A space is regular Hausdorff if it is a regular T 0 space.
2. A space is hyper-connected if no two non-empty open sets are disjoint.

### Clopen SetsUpDw('CLOPEN_SETS','-Abz-');

1. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
2. A space is zero-dimensional if it has a base of clopen sets.

### Hausdorff SpaceUpDw('HAUSDORFF_SPACE','-Abz-');

1. Therefore, every compact Hausdorff space is normal.
2. Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
3. Every normal space admits a partition of unity.
4. Every locally compact Hausdorff space is Tychonoff.

### Totally DisconnectedUpDw('TOTALLY_DISCONNECTED','-Abz-');

1. A space is totally disconnected if it has no connected subset with more than one point.
2. Every discrete space is totally disconnected.
3. Every non-empty discrete space is second category.

### Tychonoff SpaceUpDw('TYCHONOFF_SPACE','-Abz-');

1. The Stone-Čech compactification of any Tychonoff space is a compact Hausdorff space.
2. Problem 15 Let X be a cleavable (an M-cleavable) Tychonoff space.

### Totally BoundedUpDw('TOTALLY_BOUNDED','-Abz-');

1. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.
2. A space is uniformizable if and only if it is a gauge space.
3. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, still some of its subsets will be.

### Whenever SpaceUpDw('WHENEVER_SPACE','-Abz-');

1. A space X is contractible if the identity map on X is homotopic to a constant map.
2. A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.
3. A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it.

### Paracompact SpaceUpDw('PARACOMPACT_SPACE','-Abz-');

1. In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement.
2. A hereditarily paracompact space is a space such that every subspace of it is paracompact.

### CountableUpDw('COUNTABLE','-Abz-');

1. Polish. A space is called Polish if it is metrizable with a separable and complete metric.
2. A space is second-countable if it has a countable base for its topology.
3. A space is separable if it has a countable dense subset.
4. Also, any second-countable space is a Lindel--f space, but not conversely.
5. If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets.

### Topological VectorUpDw('TOPOLOGICAL_VECTOR','-Abz-');

1. This turns the dual into a locally convex topological vector space.
2. Every topological vector space has a local base of absorbing and balanced sets.

### Completely RegularUpDw('COMPLETELY_REGULAR','-Abz-');

1. Generalising both the metric spaces and the topological groups, every uniform space is completely regular.
2. Every metric space is Tychonoff; every pseudometric space is completely regular.
3. Every uniformizable space is a completely regular topological space.

### Space ParacompactUpDw('SPACE_PARACOMPACT','-Abz-');

1. A metric space is Hausdorff, also normal and paracompact.
2. Intuitively speaking, this means that the space looks the same at every point.
3. A space is paracompact if every open cover has an open locally finite refinement.
4. A space is discrete if all of its points are completely isolated, i.e.
5. A topological space is compact if and only if every filter on the space has a convergent refinement.

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

1. Information > Science > Astronomy > Universe
2. Encyclopedia of Keywords > Places > Earth > Environments
3. Information > Science > Mathematics > Topology
4. Encyclopedia of Keywords > Nature
5. Physics > Theoretical Physics > Relativity > Spacetime

### SubcategoriesUpDw('Subcategories','-Abz-');

 Astronauts (22)Rotation (1) Spacecraft (9)Space Exploration (2) Space Programs (2)Space Stations (1)
1. Books about "Space" in Amazon.com
 Short phrases about "Space"   Originally created: April 08, 2008.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
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