scholarly journals Isometries of Spaces of Radon Measures

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Marek Wójtowicz
Keyword(s):  
Closed Subset ◽  
Unit Interval ◽  
Metrizable Space ◽  
Discrete Space ◽  
Hausdorff Spaces ◽  
Bernstein Type ◽  
Radon Measures ◽  
Compact Metrizable Space

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m).

Notes on general topology

1965 ◽  
Vol 61 (4) ◽  
pp. 881-882 ◽  
Author(s):  
A. J. Ward
Keyword(s):  
Present Author ◽  
Hausdorff Space ◽  
Metrizable Space ◽  
Discrete Space ◽  
General Topology ◽  
Continuous Image ◽  
Hausdorff Spaces ◽  
Topological Limit

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.

scholarly journals Residually finite actions and crossed products

2011 ◽  
Vol 32 (5) ◽  
pp. 1585-1614 ◽  
Author(s):  
DAVID KERR ◽  
PIOTR W. NOWAK
Keyword(s):  
Free Group ◽  
Metrizable Space ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Residual Finiteness ◽  
Hausdorff Spaces ◽  
Residually Finite ◽  
Compact Metrizable Space ◽  
Continuous Actions

AbstractWe study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

On the Complete Invariance Property in Some Uncountable Products

1992 ◽  
Vol 35 (2) ◽  
pp. 221-229 ◽  
Author(s):  
Piotr Koszmider
Keyword(s):  
Fixed Point ◽  
Closed Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Unit Interval ◽  
Invariance Property ◽  
Fixed Point Set ◽  
Nonempty Closed Subset ◽  
Linear Spaces ◽  
Point Set

AbstractWe consider uncountable products of nontrivial compact, convex subsets of normed linear spaces. We show that these products do not have the complete invariance property i.e. they include a nonempty, closed subset which is not a fixed point set (i.e. the set of all fixed points) for any continuous mapping from the product into itself. In particular we give an answer to W.Weiss' question whether uncountable powers of the unit interval have the complete invariance property.

A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

1994 ◽  
Vol 37 (4) ◽  
pp. 552-555 ◽  
Author(s):  
Juris Steprans ◽  
Stephen Watson ◽  
Winfried Just
Keyword(s):  
Fixed Point ◽  
Hausdorff Space ◽  
Unique Fixed Point ◽  
Banach Fixed Point Theorem ◽  
Hausdorff Spaces ◽  
Compact Metric Space ◽  
Banach Contraction ◽  
Compact Metrizable Space ◽  
Admissible Metric ◽  
The Banach Contraction Principle

AbstractWe propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

scholarly journals A function space from a compact metrizable space to a dendrite with the hypo-graph topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Hanbiao Yang ◽  
Katsuro Sakai ◽  
Katsuhisa Koshino
Keyword(s):  
Finite Number ◽  
Function Space ◽  
Metrizable Space ◽  
Vietoris Topology ◽  
Hilbert Cube ◽  
Graph Topology ◽  
Closed Sets ◽  
Continuous Map ◽  
Compact Metrizable Space ◽  
Isolated Points

Abstract Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

scholarly journals Some classes of E-Compactness

1972 ◽  
Vol 13 (4) ◽  
pp. 492-500 ◽  
Author(s):  
Robert L. Blefko
Keyword(s):  
Closed Subset ◽  
Unit Interval ◽  
Related Question ◽  
Topological Spaces ◽  
Completely Regular ◽  
Compact Spaces

Mrowka and Engleking [1] have recently introduced a notion more general than that of compactness. Perhaps the most convenient direction at departure is the following: for spaces X and E, X is said to be E-compact if X is topologically embeddable as a closed subset of a product Em for some cardinal m, in which case we write X ⊂cl Em. More generally, X is said to be E-completely regular if X is topologically embeddable in a product Em for some m. For example, if we take E to be the unit interval I, we obtain the class of compact spaces and completely regular spaces, respectively, as is well-known. The question then arises, of course, given a space E, what spaces are compact with respect to it? A related question, to which we address ourselves in this note, is the following. Denote by K[E] all those topological spaces which are E-compact. Then we ask: are there very many distinct E-compact classes? It will develop that there are indeed quite a large number of such classes.

On Product of Radón Measures

1969 ◽  
Vol 12 (4) ◽  
pp. 427-444 ◽  
Author(s):  
M. C. Godfrey ◽  
M. Sion
Keyword(s):  
Direct Reference ◽  
Outer Measure ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Radon Measures ◽  
Closed Sets

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.

M-CHANNEL MRA AND APPLICATION TO ANISOTROPIC SOBOLEV SPACES

2005 ◽  
Vol 03 (02) ◽  
pp. 153-166
Author(s):  
ELENA CORDERO
Keyword(s):  
Tensor Product ◽  
Sobolev Spaces ◽  
Unit Interval ◽  
Biorthogonal Wavelet ◽  
Bernstein Type ◽  
Wavelet Bases ◽  
Compactly Supported ◽  
Anisotropic Sobolev Spaces ◽  
Dilation Factor ◽  
Bernstein Type Inequalities

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces.

Non-Metrizable Uniformities and Proximities on Metrizable Spaces

1973 ◽  
Vol 25 (5) ◽  
pp. 979-981
Author(s):  
P. L. Sharma
Keyword(s):  
Metrizable Space ◽  
Compact Metrizable Space ◽  
Metrizable Spaces

In the literature there exist examples of metrizable spaces admitting nonmetrizable uniformities (e.g., see [3, Problem C, p. 204]). In this paper, this phenomenon is presented more coherently by showing that every non-compact metrizable space admits at least one non-metrizable proximity and uncountably many non-metrizable uniformities. It is also proved that the finest compatible uniformity (proximity) on a non-compact non-semidiscrete space is always non-metrizable.

Reflexive objects in topological categories

1996 ◽  
Vol 6 (4) ◽  
pp. 375-386
Author(s):  
Michael D. Rice
Keyword(s):  
Geometric Structure ◽  
Unit Interval ◽  
Continuous Image ◽  
Topological Category ◽  
Hausdorff Spaces ◽  
Tychonoff Space ◽  
Countably Compact ◽  
Discrete Subspace ◽  
Cartesian Closed

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

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