Spaces of Quasi-Measures

1999 ◽  
Vol 42 (3) ◽  
pp. 291-297 ◽  
Author(s):  
D. J. Grubb ◽  
Tim LaBerge
Keyword(s):  
Direct Proof ◽  
Normal Space ◽  
Regular Space ◽  
Completely Regular

AbstractWe give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of theWheeler-Shakmatov theorem, which states that if X is normal and dim(X) ≤ 1, then every quasi-measure on X extends to a measure.

On Extensions of Topologies

1967 ◽  
Vol 19 ◽  
pp. 474-487 ◽  
Author(s):  
Carlos J. R. Borges
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Normal Space ◽  
Regular Space ◽  
Simple Extension ◽  
Completely Regular ◽  
Countably Compact

If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.

Characters on C(X)

2015 ◽  
Vol 58 (1) ◽  
pp. 7-8
Author(s):  
Karim Boulabiar
Keyword(s):  
Classical Result ◽  
Direct Proof ◽  
Regular Space ◽  
Completely Regular ◽  
Precise Condition ◽  
Large Audience

AbstractThe precise condition on a completely regular space X for every character on C(X) to be an evaluation at some point in X is that X be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

A Regular Space, Not Completely Regular

1969 ◽  
Vol 76 (2) ◽  
pp. 181 ◽  
Author(s):  
John Thomas
Keyword(s):  
Regular Space ◽  
Completely Regular

scholarly journals Intuitionistic Fuzzy Topological Spaces Model

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Hind Fadhil Abbas
Keyword(s):  
Hausdorff Space ◽  
Basic Structure ◽  
Normal Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Scientific Phenomenon ◽  
Fuzzy Topological Spaces

The fusion of technology and science is a very complex and scientific phenomenon that still carries mysteries that need to be understood. To unravel these phenomena, mathematical models are beneficial to treat different systems with unpredictable system elements. Here, the generalized intuitionistic fuzzy ideal is studied with topological space. These concepts are useful to analyze new generalized intuitionistic models. The basic structure is studied here with various relations between the generalized intuitionistic fuzzy ideals and the generalized intuitionistic fuzzy topologies. This study includes intuitionistic fuzzy topological spaces (IFS); the fundamental definitions of intuitionistic fuzzy Hausdorff space; intuitionistic fuzzy regular space; intuitionistic fuzzy normal space; intuitionistic fuzzy continuity; operations on IFS, the compactness and separation axioms.

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

scholarly journals Some results involving weak compactness in C(X), C(υX) and C(X)′

1975 ◽  
Vol 19 (3) ◽  
pp. 221-229 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Dual Space ◽  
Present Note ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Regular Space ◽  
Compact Sets ◽  
Completely Regular ◽  
Countably Compact ◽  
Countable Compactness

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

2019 ◽  
pp. 1-18
Author(s):  
Alexander J. Izzo ◽  
Dimitris Papathanasiou
Keyword(s):  
Maximal Ideal ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Regular Space ◽  
Compact Set ◽  
Gleason Part ◽  
Completely Regular ◽  
Analytic Discs ◽  
Compact Subspace

Abstract We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$ -compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to  $X$ , and $\widehat{K}$ contains no analytic discs.

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