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Topic: Locally compact space

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In the News (Tue 16 Jul 19)

  Locally compact space - Wikipedia, the free encyclopedia
The definition of a locally compact space is not universally agreed upon.
Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.
For locally compact spaces local uniform convergence is the same as compact convergence.
en.wikipedia.org /wiki/Locally_compact_space   (1365 words)

 Tychonoff space
Tychonoff spaces are named after Andrey Tychonoff[?], whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov".
Every locally compact regular space is completely regular, and every locally compact Hausdorff space is Tychonoff.
Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space.
www.ebroadcast.com.au /lookup/encyclopedia/ty/Tychonoff_space.html   (401 words)

 PlanetMath: locally compact
have a neighborhood which is actually compact, since compact open sets are fairly rare and the more relaxed condition turns out to be more useful in practice.
However, it is true that a space is locally compact at
This is version 2 of locally compact, born on 2002-05-15, modified 2004-10-24.
planetmath.org /encyclopedia/LocallyCompact.html   (101 words)

 [No title]
"Locally homogeneous" being: is connected, and for each pair of points x and y there are neighborhoods M and N and a homeomorphism f_xy: M --> N with f_xy(x) = y.
In particular, a compact space is locally compact.
If the spaces are Hausdorff, these conditions are equivalent, but in general the latter condition is stronger and it is not immediately clear that compactness implies local compactness!
www.math.niu.edu /~rusin/known-math/95/homogen   (1110 words)

 Summary   (Site not responding. Last check: 2007-11-03)
We extend the basic results on the theory of the generalized Riemann integral to the setting of bounded or locally finite measures on locally compact second countable Hausdorff spaces.
The correspondence between Borel measures on X and continuous valuations on the upper space UX gives rise to a topological embedding between the space of locally finite measures and locally finite continuous valuations, both endowed with the Scott topology.
We construct an approximating chain of simple valuations on the upper space of a locally compact space, whose least upper bound is the given locally finite measure.
www.helsinki.fi /~negri/abs-int.html   (193 words)

 directopedia : Directory : Science : Math : Topology
In that case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes of topology.
The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the doughnut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
www.directopedia.org /directory/Science-Math/Topology.shtml   (2012 words)

 Springer Online Reference Works
A topological space at every point of which there is a neighbourhood with compact closure.
The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta.
Every connected paracompact locally compact space is the sum of countably many compact subsets.
eom.springer.de /l/l060320.htm   (132 words)

 [No title]
For compact quantum groups, the appropriate differential calculi to study are covariant under the natural action of the quantum group, and "integrals" now give rise to cocycles twisted by an automorphism of the algebra.
In the case that the poset is a basis for a topological space ordered under inclusion, the fundamental group of the poset and that of the underlying topological space are isomorphic.
"Local QFT on a halfspace from non-local QFT on the boundary", Karl Henning Rehren
www.stp.dias.ie /events/2004/abstracts.html   (2271 words)

 HJM, Vol. 25, No. 2, 1999
Properties of a local near-ring with a commutative group of units are studied in the paper.
Specifically, the equivalence of finite generation for the additive group and the multiplicative group of a local near-ring is established, and a comprehensive classification of local near-rings with cyclic groups of units is given.
The local well-posedness is established for the complex Ginzburg-Landau equation with data in Sobolev spaces of negative indices.
www.math.uh.edu /~hjm/Vol25-2.html   (1495 words)

 Springer Online Reference Works
Abstract potential theory arose in the middle of the 20th century from the efforts to create a unified axiomatic method for treating a vast diversity of properties of the different potentials that are applied to solve problems of the theory of partial differential equations.
A harmonic space is locally connected, does not contain isolated points and has a basis consisting of connected resolutive sets (resolutive domains).
Deny introduced another branch of abstract potential theory: the notion of Dirichlet space, an axiomatization of the theory of the Dirichlet integral.
eom.springer.de /p/p074150.htm   (1418 words)

 Space of Ends - Equivalence of Different Definitions   (Site not responding. Last check: 2007-11-03)
            In [1] is given a description of the space of ends of a topological space by use of admissible sequences.
This description is equivalent with the general notion in the case when the topological space is locally compact, separable metric space, and its space of quasicomponents is compact.
In this paper we give a straight proof that the previous description of the space of ends of a connected, locally compact, metric space by use of admissible sequences, coincides with the description which uses inverse limit of the components of complements of compacta.
www.pmf.ukim.edu.mk /mathematics/vasilevska.htm   (212 words)

If s is locally compact and hausdorff, we can turn s into a compact space by adding the point at infinity.
The sphere is compact, hence it is the "compactification" of the plane.
The intersection of the complements of two compact sets is the complement of their union, which is another compact set.
www.mathreference.com /top-cs,cfc.html   (749 words)

 Sober spaces and continuations   (Site not responding. Last check: 2007-11-03)
A topological space is sober if it has exactly the points that are dictated by its open sets.
A new definition of sobriety is formulated in terms of lambda calculus and elementary category theory, with no reference to lattice structure, but, for topological spaces, this coincides with the standard lattice-theoretic definition.
The leading model of the new axioms is the category of locally compact locales and continuous maps.
www.tac.mta.ca /tac/volumes/10/12/10-12abs.html   (252 words)

 Local property - Wikipedia, the free encyclopedia
Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space.
Similarly, the sphere and the plane are locally equivalent.
An exception is a locally closed subset of a topological space, which is simply the intersection of an open set and a closed set.
en.wikipedia.org /wiki/Locally   (252 words)

 Topologies on Spaces of Continuous Functions (Abstract)   (Site not responding. Last check: 2007-11-03)
It is well-known that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compact-open topology.
It is less well-known that among arbitrary topological spaces, the exponentiable spaces are precisely the core-compact spaces.
Continuity of the function-evaluation map is shown to coincide with a certain approximation property of a topology on the frame of open sets of the exponent space, and the existence of a smallest approximating topology is equivalent to exponentiability of the space.
rw4.cs.uni-sb.de /~heckmann/abstracts/topfunc.html   (236 words)

 PlanetMath: weak-* topology of the space of Radon measures
PlanetMath: weak-* topology of the space of Radon measures
"weak-* topology of the space of Radon measures" is owned by stevecheng.
This is version 1 of weak-* topology of the space of Radon measures, born on 2005-07-07.
planetmath.org /encyclopedia/WeakTopologyOfTheSpaceOfRadonMeasures.html   (233 words)

 Kirk, Ronald Brian (1968-04-08) Measures in topological spaces. ...
Let X be a completely-regular topological space and let C*(X) denote the space of all bounded, real-valued continuous functions on X. For a positive linear functional [...] on C*(X), consider the following two continuity conditions.
Of particular importance is the representation of C*(X) as a space [...] of finitely-additive set functions on a certain algebra of subsets of X. This result was first announced by A. Alexandrov, but his proof was obscure.
It is shown that B-compactness is a topological invariant and various topological properties of B-compact spaces are investigated.
etd.caltech.edu /etd/available/etd-09252002-093739   (380 words)

 BGU Set Theory and Topology Seminars
Abstract: A flow is, by the definition, an action of G on a topological space X, where G is either the group of integers or the group of the real numbers; we add the adjective discrete or continuous to specify the group.
An interesting problem is which topological spaces admit minimal flows (a flow is minimal if all its orbits are dense).
Finally, I will prove that if a Hausdorff space X admits a flow whose all forward orbits are dense then X is either compact or else X is nowhere locally compact.
www.math.bgu.ac.il /~arkady/topologyseminar/topologyseminararchive.htm   (543 words)

 Baire Category Theorem
to review, a space is hausdorff if two points can be separated in disjoint open sets, and a space is locally compact if each point is contained in an open set whose closure is compact.
The common metric spaces are hausdorff and locally compact, but others are not, so this is not a generalization of the above.
The interval [0,1] is compact, hausdorff, a complete metric space, the nicest space you could think of, and the complement of any point is a dense open set, yet the intersection of these sets is empty.
www.mathreference.com /top-ms,bct.html   (621 words)

 Linear Functionals
Two linear functionals have the same kernel if and only if they are (nonzero) multiples of each other.
, the space of sequences converging to 0 with the infinity norm.
) is the vector space of all bounded linear functionals, equipped with the norm
www.math.unl.edu /~bbockelm/928/node22.html   (61 words)

 Citebase - Computably Based Locally Compact Spaces
ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus.
In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones.
Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0512110   (275 words)

 Hemicompact space - Wikipedia, the free encyclopedia
In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence.
Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.
Every first countable hemicompact space is locally compact.
en.wikipedia.org /wiki/Hemicompact_space   (107 words)

 Abstract: Stannett (1989) G-compactifications of a locally compact Hausdorff space (http://www.dcs.shef.ac.uk/~mps)   (Site not responding. Last check: 2007-11-03)
Abstract: Stannett (1989) G-compactifications of a locally compact Hausdorff space (http://www.dcs.shef.ac.uk/~mps)
Computer Science : VT Group : Mike Stannett : Research : Abstracts : G-compactifications of a locally compact Hausdorff space
G(Y) ' is equivalent to 'X is homeomorphic to Y', whenever X and Y are also compact.
www.dcs.shef.ac.uk /~mps/research/abstracts/1989G-Compactifications.html   (120 words)

 Continuous functions
The basic theory of metric spaces deals with properties of subsets (open, closed, compact, connected), sequences (convergent, Cauchy) and maps (continuous) and the relationship between these notions.
The case of a finite dimensional normed space is not very interesting because, apart from the dimension, they are all ``the same''.
This just means that every point has a compact neighborhood, i.e., is in the interior of a compact set.
www-math.mit.edu /~rbm/18.155-F02/Lecture-notes/node3.html   (885 words)

 Dual Spaces for LCS
As with Banach spaces, the dual space of a LCS
Many results from the theory of duals of Banach spaces carry over.
For example, hyperplanes are either closed or dense, a functional is continuous iff it is continuous at one point iff the kernel is closed.
www.math.unl.edu /~s-bbockel1/929/node3.html   (108 words)

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