scholarly journals The Spectral Sequence of a Covering

1961 ◽  
Vol 12 (3) ◽  
pp. 149-158 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Topological Space ◽  
Spectral Sequence ◽  
Abelian Groups ◽  
Locally Finite

Let be a covering of a topological space X and ℱ a sheaf of abelian groups over X. By a well known result of Leray, (3) II theorems 5.2.4. and 5.4.1., if is open, or closed and locally finite, there exists a spectral sequence {Er} satisfying isomorphisms and for some filtration of the graded group H*(X, ℱ). ℋq(ℱ) denotes the system of coefficients over : s→Hq(| s |, ℱ).

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

A Note on Extending Locally Finite Collections

1976 ◽  
Vol 19 (1) ◽  
pp. 117-119
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Topological Space ◽  
Set Cover ◽  
Topological Spaces ◽  
Locally Finite ◽  
Whole Space

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

scholarly journals C*-Algebras over Topological Spaces: Filtrated K-Theory

2012 ◽  
Vol 64 (2) ◽  
pp. 368-408 ◽  
Author(s):  
Ralf Meyer ◽  
Ryszard Nest
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Spectral Sequence ◽  
Topological Spaces ◽  
C Algebra ◽  
C Algebras ◽  
Finite Spaces ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Kasparov Theory

AbstractWe define the filtrated K-theory of a C*-algebra over a finite topological spaceXand explain how to construct a spectral sequence that computes the bivariant Kasparov theory overXin terms of filtrated K-theory.For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a spaceXwith four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this spaceX, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

scholarly journals The skeleton of a variety of groups

1972 ◽  
Vol 6 (3) ◽  
pp. 357-378 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Product Variety ◽  
Finite Variety ◽  
Locally Finite ◽  
Variety Of Groups ◽  
Locally Finite Variety ◽  
Countably Infinite ◽  
Relatively Free Group

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

scholarly journals Remarks on γˆ*Ω-open sets and minimal γˆ*Ω-open sets γˆ*Ω

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1017-1021
Keyword(s):  
Topological Space ◽  
Locally Finite ◽  
Basic Properties ◽  
Finite Space ◽  
The Impact

Aim of this paper is to define -open sets in a topological space and obtain their basic properties. Also, we define minimal -open sets in a space and study the impact of two minimal -open sets in a space with -regular operation. However, the roll of minimal -open sets in -locally finite space has been discussed.

scholarly journals Bombay hypertopologies

2003 ◽  
Vol 4 (2) ◽  
pp. 421 ◽  
Author(s):  
Giuseppe Di Maio ◽  
Enrico Meccariello ◽  
Somashekhar Naimpally
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Wijsman Convergence ◽  
Locally Finite ◽  
Simple Result ◽  
Special Cases ◽  
Wijsman Topology

<p>Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.</p>

A Spectral Sequence for Cohomotopy

1969 ◽  
Vol 21 ◽  
pp. 712-729 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Abelian Group ◽  
Spectral Sequence ◽  
Prime Number ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Image Position

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by

On Groups Saturated with Abelian Subgroups

1998 ◽  
Vol 08 (04) ◽  
pp. 443-466 ◽  
Author(s):  
Lev S. Kazarin ◽  
Leonid A. Kurdachenko ◽  
Igor Ya. Subbotin
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Normal Subgroups ◽  
Main Subject ◽  
Maximal Condition ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Locally Nilpotent ◽  
Abelian Subgroups ◽  
Weak Maximal Condition

Groups with the weak maximal condition on non-abelian subgroups are the main subject of this research. Locally finite groups with this property are abelian or Chemikov. Non-abelian groups with the weak maximal condition on non-abelian subgroups, which have an ascending series of normal subgroups with locally nilpotent or locally finite factors, are described in this article.

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