scholarly journals A topological zero-one law for open continuous maps

1988 ◽  
Vol 38 (2) ◽  
pp. 263-266
Author(s):  
E. Barone ◽  
K.P.S. Bhaskara Rao
Keyword(s):  
Topological Space ◽  
Baire Property ◽  
Continuous Maps

We obtain a topological zero-one law for sets with the Baire property which are invariant under a semigroup of open continuous maps acting on a topological space.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Classifying Algebras for the K-Theory of σ-C*-Algebras

1989 ◽  
Vol 41 (6) ◽  
pp. 1021-1089 ◽  
Author(s):  
N. Christopher Phillips
Keyword(s):  
Hilbert Space ◽  
Topological Space ◽  
Fredholm Operators ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Continuous Maps ◽  
Group Structures

In topology, the representable K-theory of a topological space X is defined by the formulas RK0(X) = [X,Z x BU] and RKl(X) = [X, U], where square brackets denote sets of homotopy classes of continuous maps, is the infinite unitary group, and BU is a classifying space for U. (Note that ZxBU is homotopy equivalent to the space of Fredholm operators on a separable infinite-dimensional Hilbert space.) These sets of homotopy classes are made into abelian groups by using the H-group structures on Z x BU and U. In this paper, we give analogous formulas for the representable K-theory for α-C*-algebras defined in [20].

scholarly journals Relations of Pre Generalized Regular Weakly Locally Closed Sets in Topological Spaces

2021 ◽  
Vol 12 (3) ◽  
pp. 3444-3454
Author(s):  
Vijayakumari T Et.al
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Set ◽  
Closed Sets ◽  
Open Set ◽  
Continuous Maps ◽  
Locally Closed Sets

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

Associated Regular Spaces

1968 ◽  
Vol 20 ◽  
pp. 1087-1092 ◽  
Author(s):  
J. Pelham Thomas
Keyword(s):  
Topological Space ◽  
Regular Space ◽  
Continuous Maps ◽  
Regular Topology

Let be any topological space. In this paper, we show that there is a unique regular topology on X which is coarser than such that if Y is any regular space, the continuous maps are the same for and . We shall call the regular topology associated with and the regular space associated with .

On the Complete Regularity of Some Category Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 651-656 ◽  
Author(s):  
W. Eames
Keyword(s):  
Topological Space ◽  
Density Function ◽  
Measure Space ◽  
Baire Property ◽  
Complete Regularity ◽  
Completely Regular ◽  
Upper Density ◽  
Covering Class ◽  
Sequential Covering ◽  
Null Set

A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

scholarly journals On Almost-Fixed-Point Theory

1978 ◽  
Vol 30 (4) ◽  
pp. 673-699 ◽  
Author(s):  
Michiel Hazewinkel ◽  
Marcel Van De Vel
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Finite Covering ◽  
Continuous Maps

Let X be a topological space, a finite covering of X (the words ‘covering’ and ‘cover’ are used interchangeably). We say that has the almost fixed point property for a class of continuous maps f : X → X if for all there is an x ∈ X and such that x ∈ U and f(x) ∈ U, or, equivalently, if there is a such that .

scholarly journals On Perturbations of Continuous Maps

2013 ◽  
Vol 56 (1) ◽  
pp. 92-101
Author(s):  
Benoît Jacob
Keyword(s):  
Perturbation Theory ◽  
Topological Space ◽  
Small Perturbation ◽  
Sufficient Conditions ◽  
Matrix Perturbation ◽  
Continuous Map ◽  
Matrix Perturbation Theory ◽  
Continuous Maps

AbstractWe give sufficient conditions for the following problem: given a topological space X, ametric space Y, a subspace Z of Y, and a continuous map f from X to Y, is it possible, by applying to f an arbitrarily small perturbation, to ensure that f(X) does not meet Z? We also give a relative variant: if f(X') does not meet Z for a certain subset X'⊂ X, then we may keep f unchanged on X'. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory.

scholarly journals On the regularity of topologies in the family of sets having the Baire property

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1291-1295 ◽  
Author(s):  
Jacek Hejduk
Keyword(s):  
Topological Space ◽  
Regularity Property ◽  
Baire Property ◽  
Proper Ideal ◽  
The Family ◽  
Meager Sets

The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager sets. The regularity property of such topologies is investigated.

An application of descent to a classification theorem for toposes

1990 ◽  
Vol 107 (1) ◽  
pp. 59-79 ◽  
Author(s):  
Marta Bunge
Keyword(s):  
Topological Space ◽  
Discrete Group ◽  
Classification Theorem ◽  
Isomorphism Classes ◽  
Continuous Maps ◽  
Geometric Morphisms

The aim of this paper is to answer the following question. For a spatial groupoid G, i.e. for a groupoid in the category Sp of spaces (in the sense of [20]) in a topos , and continuous maps, the topos BG, of étale G-spaces, is called ‘the classifying topos of G’ by Moerdijk[22]. This terminology is suggested by the case of G a discrete group (in Sets), as then BG, the topos of G-sets, classifies principal G-bundles. This means that, for each topological space X, there is a bijection between isomorphism classes of principal G-bundles over X and isomorphism classes of geometric morphisms from Sh(X) to BG. The question is: what does BG classify, in terms of G, in the general case of a spatial groupoid G in a topos ?

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