# topological groups

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## topological groups

[¦täp·ə¦läj·ə·kəl ′grüps]
(mathematics)
Groups which also have a topology with the property that the group operation and the inverse operation determine continuous functions.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Graev, "Free topological groups," Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, vol.
When S is a topological group, then LUC(S) is the set of all uniformly continuous functions on S with respect to the right uniformity of S i.e., f [member of] LUC(S) [??] [for all] [epsilon] > 0, [there exists] U neighborhood of the identity of S such that [s.sup.-1]t [member of] U [??] [absolute value of f (s) - f (t)] [less than or equal to] [epsilon].
If we set S := G x T, where G is an abelian topological group, then S is a reflexive foundation semigroup and again by Theorem 4.1, [M.sub.a](S) is BSE.
Keywords: Semi open set, semi closed set, irresolute mapping, semi homeomorphism, irresolute topological group, semi connected space, semi component, semi topological groups with respect to irresoluteness.
In studying the strong topologies on the generalized PN spaces, Alsina et al., 1997 [3] investigated the continuity of the probabilistic norm; they pointed out that each PN space is a topological group but may not be a topological vector space.
Among the topics are functional algebras of operators generated by a self-adjoint operator in Pontryagin space ?1, Wedderburn structure theorems for two-sided locally m-convex H*-algebras, main embedding theorems for symmetric spaces of measurable functions, discrete non-closed subsets in maximally non-discrete topological groups, faithfully representable topological *-algebras: some spectral properties, and dense ideals in topological algebras: some results and open problems.
By definition of [bar.[??]], we have the following commutative diagram of (compact) topological groups with exact rows
As it has been done for topological groups; we wish to introduce a topol- ogy on hypergroup in such a way that the hyperoperation becomes continue.
Singh also studies symmetric continuous cohomology of topological groups in [1].
* A theorem on functional Alexandroff topological groups.
The 64 papers in this collection explore field theory and polynomials, commutative rings and algebras, matrix theory, associative rings, K-theory, group theory and generalizations, topological groups, Lie groups, and differential geometry.

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