Symplectic space (152 bytes)
1: ...ace''' is either a [[symplectic manifold]] or a [[symplectic vector space]]. The latter is a special case of t...

Symplectic matrix (2125 bytes)
1: In [[mathematics]], a '''symplectic matrix''' is a ''2n'' by ''2n'' [[matrix (mathema...
12: Every symplectic matrix has an [[inverse matrix|inverse]] which is...
14: ...[Lie group]] called the [[symplectic group]]. The symplectic group has dimension ''n''(2''n'' + 1).
16: ...om the definition that the [[determinant]] of any symplectic matrix is ±1. Actually, it turns out that ...
22: ...'n'' matrices. Then the condition for ''M'' to be symplectic is equivalent to the conditions

Symplectic group (5629 bytes)
1: ...'). The latter is sometimes called the '''compact symplectic group''' to distinguish it from the former. Note ...
5: ...ymplectic matrices have unit [[determinant]], the symplectic group is a [[subgroup]] of the [[special linear g...
7: ...or space]]''. The symplectic group of an abstract symplectic vector space ''V'' is also denoted Sp(''V'').
9: When ''n'' = 1, the symplectic condition on a matrix is satisfied [[iff]] the de...
25: The symplectic group, Sp(''n''), is the subgroup of GL(n, '''H''...

Symplectic Geometry (4065 bytes)
1: In [[geometry]], ''Symplectic Geometry'' is the study of vector spaces and mani...
3: === Symplectic Vector Spaces ===
5: A symplectic form on a [[vector space]] <math>V</math> over a ...
11: ...ic vector space is one that comes equipped with a symplectic form.
15: ...hmidt process]] shows that any finite-dimensional symplectic vector space <math>V</math> is isomorphic to <mat...

Symplectic representation (2264 bytes)
1: ...ebra]] over the [[real number]]s, can exist. A '''symplectic representation''' is a particular kind of pseudor...
3: ...f a [[direct sum]] of a real representation and a symplectic representation, as satisfying the stated conditio...
5: ...ble group representation that is neither real nor symplectic is a [[complex representation]].
7: ... for the representation space, there exists for a symplectic represention an [[antilinear map]]

Symplectic manifold (7590 bytes)
1: ...he study of symplectic manifolds is '''symplectic topology'''.
5: Symplectic manifolds arise naturally in abstract formulation...
9: ...n example of a ''linear'' symplectic space. See [[symplectic vector space]].
13: ...''symplectic volume form''. It follows that every symplectic manifold is canonically [[oriented]] and comes wi...
17: Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorph...

Symplectic vector space (3191 bytes)
1: ...metric]], [[bilinear form]] &omega; called the '''symplectic form'''.
3: Explicitly, a symplectic form is a bilinear form &omega; : ''V'' &times; '...
6: ... (Note that this is ''not'' the same thing as a [[symplectic matrix]].)
10: ==Standard symplectic space==
12: ...lectic space is '''R'''^2''n'' with the symplectic form given by the [[block matrix]]

Topology (11069 bytes)
3: '''Topology''' ([[Greek language|Greek]] ''topos'' = place an...
5: Topology is concerned with the study of the so-called [[to...
14: ...y]] contains definitions of terms used throughout topology.
17: The root of topology was in the study of geometry in ancient cultures....
29: ...e status of the '''topologist''', a specialist in topology, was probably established from around 1920.

Point (topology) (160 bytes)
1: [[Category:Topology]]
2: In [[mathematics]], particularly in [[topology]], a [[point]] of a [[topological space]] is an e...

Strong topology (472 bytes)
1: ...ogy]] which is stronger than some other "default" topology. This term is used to describe different topologi...
2: * the topology arising from a [[normed vector space|norm]]
3: * the [[strong operator topology]]
5: ...is stronger than a topology &sigma; (is a [[finer topology]]) if it contains more [[open set]]s.
8: * [[weak topology]]

Alexandrov topology (13413 bytes)
1: In [[general topology]] the open sets of a [[topological space]] satisf...
6: ...sets is open?"'' The answer is, the '''Alexandrov topology'''.
29: ...ization preorder]] of '''''X''''' i.e. the finest topology giving the preorder &le; satisfying ''x'' &le; ''...
37: ...subspace characterization.''' ''T'' is the finest topology consistent with the topologies of the finite subs...
41: ... their topology ''T'' is called the '''Alexandrov topology''', named after the Russian mathematician [[Pavel...

Continuity (topology) (3981 bytes)
1: [[Category:Topology]][[Category:General topology]]
2: In [[topology]], a '''continuous''' function is generally defin...
9: ...e <math>x \in X</math> if for any [[neighborhood (topology)|neighborhood]] <var>V</var> of <var>f</var>(<var...
11: <center>[[image:continuity_topology.png|Continuity of a function at a point]]</center...
29: ...uous. If the domain set is given the [[indiscrete topology]] and the range set is at least [[T0 space|T<sub>...

Combinatorial topology (720 bytes)
1: ...ial topology''' was an older name for [[algebraic topology]], dating from the time when [[Topological invari...
5: [[Category:Algebraic topology]]

Trivial topology (4130 bytes)
1: ...gy]], a [[topological space]] with the '''trivial topology''' is one where the only [[open set]]s are the [[...
3: ...r> with more than [[one]] element and the trivial topology lacks a key desirable property: it is not a [[T0 ...
8: * The only possible [[basis (topology)|basis]] of <var>X</var> is {<var>X</var>}.
11: * Not being Hausdorff, ''X'' is not an [[order topology]], nor is it [[metrizable]].
16: ...ogy)|subspace]]s of <var>X</var> have the trivial topology.

Finer topology (1785 bytes)
1: ...oarser''' (alt. '''weaker''' or '''smaller''') '''topology''' than &tau;₁.
5: ...>) to (''X'',&tau;₂), is [[continuity (topology)|continuous]]. If &tau;₁ is the finer ...
7: ... The coarsest topology on ''X'' is the [[trivial topology]]. Any two topologies on ''X'' have a ''meet'' an...
12: *[[Weak topology]]
13: *[[Strong topology]]

Geometric topology (1821 bytes)
1: ...e to be almost synonymous with '''low-dimensional topology''', concerning in particular objects of two, thre...
3: ...l technique. These fields all rested on [[general topology]], which was the study of the general [[topologic...
5: The methods of geometric topology being combinatorial, and difficulties arising (fo...
7: ...ainly broke down the barriers with other parts of topology and geometry, making the subject more part of the...
9: See also: [[list of geometric topology topics]]

Fork (topology) (1319 bytes)
1: ...aracterization of [[graph]]s, including [[network topology]], and [[topological space]]s.
5: ...e [[Glossary of graph theory#Graph topology|graph topology]] of a graph with a fork.
6: Stated in terms of [[topology]] alone, a topological space ''X'' has a fork if ...

Upper topology (246 bytes)
1: ...e '''upper topology''' is the [[topological space|topology]] defined on a [[preorder|preordered set]], in wh...
3: [[Category:General topology]]

Signature (topology) (1596 bytes)
23: ...integral quadratic form]] are also of interest in topology.
27: [[Category:Geometric topology]][[Category:Quadratic forms]]

Path (topology) (3430 bytes)
1: ... a [[topological space]] ''X'' is a [[continuous (topology)|continuous map]] ''f'' from the [[unit interval]...
18: ... are extremely important in branch of [[algebraic topology]] called [[homotopy theory]]. A [[homotopy]] of p...
29: [[Category:Topology]]

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