scholarly journals Minimal Hausdorff and maximal compact spaces

1963 ◽  
Vol 3 (2) ◽  
pp. 167-171 ◽  
Author(s):  
N. Smythe ◽  
C. A. Wilkins
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Hausdorff Topology ◽  
Compact Spaces ◽  
Compact Topology ◽  
Non Hausdorff Spaces

Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.

scholarly journals Skew compact semigroups

2003 ◽  
Vol 4 (1) ◽  
pp. 133
Author(s):  
Ralph D. Kopperman ◽  
Desmond Robbie
Keyword(s):  
Continuous Semigroup ◽  
Minimal Ideal ◽  
Compact Semigroup ◽  
Closed Ideal ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Compact Semigroups ◽  
Compact Topology ◽  
Hausdorff Group ◽  
Non Hausdorff Spaces

<p>Skew compact spaces are the best behaving generalization of compact Hausdorff spaces to non-Hausdorff spaces. They are those (X ; τ ) such that there is another topology τ* on X for which τ V τ* is compact and (X; τ ; τ*) is pairwise Hausdorff; under these conditions, τ uniquely determines τ *, and (X; τ*) is also skew compact. Much of the theory of compact T<sub>2</sub> semigroups extends to this wider class. We show:</p> <p>A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ<sup>2</sup>→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.</p> <p>A skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de Groot semigroup. Given one of these, we show:</p> <p>It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S<sup>2</sup> = S.</p> <p>Its topology arises from its subinvariant quasimetrics.</p> <p>Each *-closed ideal ≠ S is contained in a proper open ideal.</p>

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

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.

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)).

A Banach–Stone theorem for spaces of weak* continuous functions

1985 ◽  
Vol 101 (3-4) ◽  
pp. 203-206 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Hausdorff Spaces ◽  
Space Of Continuous Functions ◽  
Dual Banach Space ◽  
Stone Theorem

SynopsisIf X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

Extension Closed and Cluster Closed Subspaces

1972 ◽  
Vol 24 (6) ◽  
pp. 1132-1136 ◽  
Author(s):  
Douglas Harris
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Closure Properties ◽  
Compact Spaces ◽  
Fixed Space

One of the most useful properties of a compact Hausdorff space is that such a space is closed whenever embedded into a Hausdorff space. This property does not extend to compact spaces with respect to embeddings into arbitrary spaces. Thus, an interesting topological problem is to characterize the types of absolute “closure” properties that are possessed by compact spaces. This is the problem that is solved in the present paper.The following notation and terminology will be used below. We shall consider a fixed space X and subspace A, representing arbitrary nonempty open subsets of X (respectively A ) by W (respectively V).

scholarly journals Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1431-1437
Author(s):  
Jeremy Siegert
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Covering Dimension ◽  
Polish Spaces ◽  
Inductive Dimension ◽  
Isolated Points

We show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer (for Polish spaces without isolated points) on the class of compact Hausdorff spaces. This shows that Fedorchuk?s example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov. This answers Isbell?s question of whether or not proximity inductive dimension and proximity dimension coincide.

scholarly journals Dynamic asymptotic dimension for actions of virtually cyclic groups

2021 ◽  
pp. 1-9
Author(s):  
Massoud Amini ◽  
Kang Li ◽  
Damian Sawicki ◽  
Ali Shakibazadeh
Keyword(s):  
Cyclic Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Direct Consequence ◽  
Asymptotic Dimension ◽  
Hausdorff Spaces ◽  
Cyclic Groups ◽  
Finite Dimensional ◽  
Famous Result ◽  
Free Actions

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$ -algebras arising from minimal free $\mathbb {Z}$ -actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Alexandroff Theorem in Hausdorff Topology for Null-Null-Additive Set Multifunctions

2013 ◽  
Vol 59 (2) ◽  
pp. 237-251 ◽  
Author(s):  
Alina Cristiana Gavriluţ
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Type Theorem ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Locally Compact Hausdorff Space

Abstract In this paper we further a previous study concerning abstract regularity for monotone set multifunctions, with has immediate applications in well-known situations such as the Borel δ-algebra of a Hausdorff space and/or the Borel (Baire, respectively) δ-ring or δ-ring of a locally compact Hausdorff space. We also study relationships among abstract regularities and other properties of continuity. Especially, a set-valued Alexandroff type theorem is obtained.

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