Topological ring (2833 bytes)
1: ...a [[ring (algebra)|ring]] ''R'' which is also a [[topological space]] such that both the addition and the multi...
9: ...al rings. See [[hypercomplex numbers]] for higher dimensional examples.
11: ...''n''</sup> &sube; ''U''. This turns ''R'' into a topological ring. The ''I''-adic topology is [[Hausdorff spa...
17: ...ed'': one can find an essentially unique complete topological ring ''S'' which contains ''R'' as a [[dense (top...
20: ... most naturally defined as completions of certain topological rings carrying ''I''-adic topologies.

Topological defect (2584 bytes)
1: ...own in regions that spread at the speed of light; topological defects occur where different regions came into c...
3: Various different types of topological defects are possible with the type of defect form...
5: *[[Domain wall]]s, two-dimensional membranes that form when a discrete symmetry is...
6: *[[Cosmic string]]s, one-dimensional lines that form when an axial or cylindrical sy...
10: ...n artificial Earth-bound physics experiments, but topological defects that formed during the universe's formati...

Topological sorting (2032 bytes)
1: ...re all nodes to which it has edges. Any DAG has a topological sort, and in fact most have many.
5: ...e completed before job ''y'' can be done. Then, a topological sort gives an order in which to perform the jobs....
9: | The graph shown to the left has many valid topological sorts, including:
18: The usual algorithm for topological sorting has running time linear in the number of ...
32: ...NIST Dictionary of Algorithms and Data Structure: topological sort]

Soliton (topological) (1115 bytes)
1: A '''topological [[soliton]]''' is a solution of a system of [[par...
3: ...cording to Kibble-Higgs mechanism. The well known topological defects are [[magnetic monopoles]], [[cosmic stri...

Topological group (6068 bytes)
1: ...groups are [[group object]]s in the [[category of topological spaces]]). Though we do not do so here, it is com...
3: ...tigated in [[mathematical analysis|analysis]] are topological groups (usually with some additional structure).
7: ...[[discrete group]]s. In this sense, the theory of topological groups subsumes that of ordinary groups.
9: ...h as [[Banach space]]s or [[Hilbert space]]s, are topological groups.
11: ...)|matrices]] with real entries can be viewed as a topological group with the topology defined by viewing GL(''n...

Topological property (584 bytes)
1: ... '''topological invariant''' is a property of a [[topological space]] which is [[invariant_(mathematics)|invari...
3: ==Topological properties==

Topological space (19036 bytes)
1: ...ng notion. The branch of mathematics that studies topological spaces in their own right is called [[topology]].
5: A '''topological space''' is a [[set]] ''X'' together with a colle...
15: ...ologies can be placed on nearly any set to form a topological space. When every set in a topology ''T''<sub>1</...
19: A [[function (mathematics)|function]] between topological spaces is said to be '''[[continuity (topology)|c...
21: ...category of topological spaces]], '''Top''', with topological spaces as [[object (category theory)|objects]] an...

Topological abelian group (674 bytes)
1: ...opological abelian group''', or '''TAG''', is a [[topological group]] that is also an [[abelian group]].
2: ...a TAG is both a [[group (algebra)|group]] and a [[topological space]], the group operations are [[continuous (t...
4: The theory of topological groups applies also to TAGs, but more can be done...
9: [[Category:Topological groups]]

Category of topological spaces (2477 bytes)
1: ...us. The study of '''Top''' and of properties of [[topological space]]s using the techniques of [[category theor...
3: ...e name '''Top''' for the category with [[Manifold|topological manifolds]] as objects and continuous maps as mor...
7: ...Top'''; any [[singleton (mathematics)|singleton]] topological space is a [[terminal object]]. There are thus no...
9: ...coproduct]] is given by the [[disjoint union]] of topological spaces. By using the [[quotient topology]], one c...
13: ...ategory of sets|'''Set''']] which assigns to each topological space the underlying [[set]], and to each continu...

Topological Boolean algebra (548 bytes)
1: ...abstract algebra]] and [[mathematical logic]], '''topological Boolean algebra''' is one of the many names that ...
3: ...)|join]] and [[Lattice (order)|complement]] are [[topological space|continuous functions]].

Topological vector space (7707 bytes)
1: In [[mathematics]], a '''topological vector space''' ''X'' is a [[real number|real]] o...
3: ...gated in [[functional analysis]]. The elements of topological vector spaces are typically [[function (mathemati...
7: ... space]]s and [[Hilbert space]]s) are examples of topological vector spaces.
9: ...t topology]]. With this topology, ''X'' becomes a topological vector space, called the ''space of pointwise con...
12: ...erentiable functions on ''D'' is a locally convex topological vector space.

Topological quantum field theory (409 bytes)
1: '''Topological quantum field theory''' in [[theoretical physics]...

Locally convex topological vector space (1481 bytes)
1: In [[functional analysis]], a [[topological vector space]] is called
5: ...r space''' (or '''locally convex space''') is a [[topological vector space]] with the following ''local convexi...
9: ... topological vector spaces are described in the [[topological vector space]] article. On the other hand, [[Lp ...

Fifth Dimension (1764 bytes)
1: '''The 5th Dimension''' is an [[United States| American]] [[popular mu...
10: Some of the songwriters popularised by The 5th Dimension later went on to [[careers]] of their own, especi...
23: *''The 5th Dimension/Live!'' (1971)
38: ...album]] called '''[[Fifth Dimension_(album)|Fifth Dimension]]'''.

VC dimension (2857 bytes)
1: ...he '''VC dimension''' (for '''Vapnik Chervonenkis dimension''') is a measure of the [[capacity]] of a [[class...
7: ...tion of capacity, called the VC dimension. The VC dimension of a model <math>f</math> is the maximum <math>h<...
9: The VC dimension has utility in statistical learning theory, becau...
15: ...ath>1-\eta</math>, where <math>h</math> is the VC dimension of the classification model, and <math>N</math> i...
19: ...p://www-2.cs.cmu.edu/~awm/tutorials/vcdim.html VC dimension tutorial]

La Dimension X (176 bytes)
1: '''La Dimension X''' is a [[French science fiction]] [[novel]] wr...

Dimension X (785 bytes)
1: '''''Dimension X''''' was an [[old-time radio]] program broadcas...
3: Dimension X was the first notable adult [[science fiction]]...
8: Dimension X lasted only one year, but the later [[X Minus O...

Dimension 5 (314 bytes)
1: '''Dimension 5''' are [[Kerry Palmer]], [[Nick Wenham]], [[Jud...
5: *[[Transdimensional]] ([[Intastella Records]] [[1997]])

Dimension (6632 bytes)
2: ...ticle is about. Also, in [[science fiction]], a "dimension" can also refer to a separate world or plane of e...
4: ==Physical dimensions==
6: ...irely sensible) to consider this as three spatial dimensions and one of time. We can move up-or-down, north-o...
8: ...n'. It is somewhat different to the three spatial dimensions in that there is only one of it, and movement se...
10: ...that the universe measured along these additional dimensions is subatomic in size.

Isoperimetric dimension (6407 bytes)
1: ... the [[topological dimension]] or the [[Hausdorff dimension]] which compare different ''local behaviors'' aga...
3: ...face area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is ''...
7: ... about a manifold ''M'' that it satisfies a ''d''-dimensional '''isoperimetric inequality''' if for any open ...
11: ...dimensional volume and area refers to (''n'' - 1)-dimensional volume. ''C'' here refers to some constant, whi...
13: ... on all ''d''-s such that ''M'' satisfies a ''d''-dimensional isoperimetric inequality.

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