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 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).

Projective Elements in Categories with Perfect θ-Continuous Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 872-884 ◽  
Author(s):  
Hans Vermeer ◽  
Evert Wattel
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
List Type ◽  
Hausdorff Spaces ◽  
Extremally Disconnected ◽  
Continuous Maps

In 1958 Gleason [6] proved the following :THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces:(i) The category of Tychonov spaces and perfect continuous functions. [4] [11].(ii) The category of regular spaces and perfect continuous functions. [4] [12].(iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1].(iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14].In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps.

Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

1973 ◽  
Vol 16 (3) ◽  
pp. 435-437 ◽  
Author(s):  
C. Eberhart ◽  
J. B. Fugate ◽  
L. Mohler
Keyword(s):  
Hausdorff Space ◽  
Disjoint Union ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
The One ◽  
One Point Compactification ◽  
Compact Subsets

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

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.

Simple Links in Locally Compact Connected Hausdorff Spaces are Nondegenerate

1981 ◽  
Vol 34 (2) ◽  
pp. 349-355
Author(s):  
David John
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Connected Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Peano Continua

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

scholarly journals On projective lift and orbit spaces

1994 ◽  
Vol 50 (3) ◽  
pp. 445-449 ◽  
Author(s):  
T.K. Das
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covariant Functor ◽  
Hausdorff Spaces ◽  
Orbit Spaces ◽  
Locally Compact ◽  
Discontinuous Group ◽  
Coreflective Subcategory ◽  
Group Of Homeomorphisms ◽  
Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

Semi-Hausdorff Spaces

1966 ◽  
Vol 9 (3) ◽  
pp. 353-356 ◽  
Author(s):  
M.G. Murdeshwar ◽  
S.A. Naimpally
Keyword(s):  
Hausdorff Space ◽  
Hausdorff Spaces

It is well-known that in a Hausdorff space, a sequence has at most one limit, but that the converse is not true. The condition that every sequence have at most one limit will be called the semi-Hausdorff condition. We will prove that the semi-Hausdorff condition is strictly stronger than the T1 -axiom and is thus between the T1 and T2 axioms. In this note, we investigate into some properties of the spaces satisfying the semi-Hausdorff condition.

scholarly journals On dimension andweight of a local contact algebra

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5481-5500
Author(s):  
G. Dimov ◽  
E. Ivanova-Dimova ◽  
I. Düntsch
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Local Contact ◽  
Continuous Maps ◽  
Contact Algebra ◽  
Contact Algebras ◽  
Locally Compact Hausdorff Space

As proved in [16], there exists a duality ?t between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X) = wa(?t(X)), and if, in addition, X is normal, then dim(X) = dima(?t(X)).

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.

scholarly journals Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Manuel Felipe Cerpa-Torres ◽  
Michael A. Rincón-Villamizar
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Geometrical Parameter ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Continuous Image ◽  
Topological Properties ◽  
Locally Compact ◽  
Regular Subspace ◽  
Locally Compact Hausdorff Space

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

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