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 The Centre of the Spaces of Banach Lattice-Valued Continuous Functions on the Generalized Alexandroff Duplicate

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Faruk Polat
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Isolated Points ◽  
Discrete Functions

We characterize the centre of the Banach lattice of Banach lattice -valued continuous functions on the Alexandroff duplicate of a compact Hausdorff space in terms of the centre of , the space of -valued continuous functions on . We also identify the centre of whose elements are the sums of -valued continuous and discrete functions defined on a compact Hausdorff space without isolated points, which was given by Alpay and Ercan (2000).

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)).

scholarly journals Double-dual types over the Banach spaceC(K)

2005 ◽  
Vol 2005 (16) ◽  
pp. 2533-2545
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Isolated Points

LetKbe a compact Hausdorff space andC(K)the Banach space of all real-valued continuous functions onK, with the sup-norm. Types overC(K)(in the sense of Krivine and Maurey) can be uniquely represented by pairs(ℓ,u)of bounded real-valued functions onK, whereℓis lower semicontinuous,uis upper semicontinuous,ℓ≤u, andℓ(x)=u(x)for all isolated pointsxofK. A condition that characterizes the pairs(ℓ,u)that represent double-dual types overC(K)is given.

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 A note on the lattice of density preserving maps

2005 ◽  
Vol 72 (1) ◽  
pp. 1-6
Author(s):  
Sejal Shah ◽  
T.K. Das
Keyword(s):  
Complete Lattice ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Continuous Maps ◽  
Countably Compact ◽  
Isolated Points ◽  
Locally Compact Hausdorff Space

We study here the poset DP (X) of density preserving continuous maps defined on a Hausdorff sapce X and show that it is a complete lattice for a compact Hausdorff space without isolated points. We further show that for countably compact T3 spaces X and Y without isolated points, DP (X) and DP (Y) are order isomorphic if and only if X and Y are homeomorphic. Finally, Magill's result on the remainder of a locally compact Hausdorff space is deduced from the relation of DP (X) with posets IP (X) of covering maps and EK (X) of compactifications respectively.

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.

scholarly journals On vector lattice-valued measures II

1986 ◽  
Vol 40 (2) ◽  
pp. 234-252 ◽  
Author(s):  
T. V. Panchapagesan ◽  
Shivappa Veerappa Palled
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Vector Lattice ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Baire Measure ◽  
Strongly Regular ◽  
Locally Compact Hausdorff Space

AbstractFor a weakly (, )-distributive vector lattice V, it is proved that a V {}-valued Baire measure 0 on a locally compact Hausdorff space T admits uniquely regular Borel and weakly Borel extensions on T if and only if 0 is strongly regular at . Consequently, for such a vector lattice V every V-valued Baire measure on a locally compact Hausdorff space T has unique regular Borel and weakly Borel extensions. Finally some characterisations of a weakly (, )-distributive vector lattice are given in terms of the existence of regular Borel (weakly Borel) extensions of certain V {}-valued Barie measures on locally compact Hausdorff spaces.

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 Locally trivial W∗-bundles

2016 ◽  
Vol 27 (11) ◽  
pp. 1650088
Author(s):  
Samuel Evington ◽  
Ulrich Pennig
Keyword(s):  
Automorphism Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covering Dimension

We prove that a tracially continuous W[Formula: see text]-bundle [Formula: see text] over a compact Hausdorff space [Formula: see text] with all fibers isomorphic to the hyperfinite II1 factor [Formula: see text] that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group [Formula: see text] shown by Popa and Takesaki. There is no restriction on the covering dimension of [Formula: see text].

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