A Regular Space, Not Completely Regular

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

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.

scholarly journals Invariant metrics on free topological groups

1973 ◽  
Vol 9 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Sidney A. Morris ◽  
H.B. Thompson
Keyword(s):  
Topological Group ◽  
Invariant Metrics ◽  
Regular Space ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Totally Disconnected

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

scholarly journals On -Cogenerated Commutative Unital -Algebras

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Ehsan Momtahan
Keyword(s):  
Compact Space ◽  
Commutative Algebra ◽  
Cardinal Number ◽  
Continuous Functions ◽  
Regular Space ◽  
Completely Regular ◽  
Simple Characterization ◽  
Maximal Ideals ◽  
Complex Valued

Gelfand-Naimark's theorem states that every commutative -algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space , is dense--separable if and only if is -cogenerated if and only if every family of maximal ideals of with zero intersection has a subfamily with cardinal number less than and zero intersection. This gives a simple characterization of -cogenerated commutative unital -algebras via their maximal ideals.

scholarly journals Weakly compact operators and the strict topologies

1989 ◽  
Vol 39 (3) ◽  
pp. 353-359 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

Liftings for functions with values in a completely regular space

1970 ◽  
Vol 187 (3) ◽  
pp. 200-206 ◽  
Author(s):  
C. Ionescu Tulcea
Keyword(s):  
Regular Space ◽  
Completely Regular
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