scholarly journals A Functional Analytic Description of Normal Spaces

1972 ◽  
Vol 24 (1) ◽  
pp. 45-49 ◽  
Author(s):  
E. Binz ◽  
W. Feldman
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Closed Ideal ◽  
Analytic Description ◽  
Completely Regular ◽  
Hausdorff Topological Space ◽  
Continuous Convergence ◽  
Convergence Structure

Throughout the paper, X will denote a completely regular (Hausdorff) topological space and C(X) the R-algebra of all real-valued continuous functions on X. When this algebra carries the continuous convergence structure [1], we write CC(X). We note that CC(X)is a complete [5] convergence R-algebra [1].Our description of normality reads as follows. A completely regular topological space X is normal if and only if CC(X)/J (endowed with the obvious quotient structure; see § 1) is complete for every closed ideal J ⊂ CC(X).

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.

scholarly journals Characterizations of Ideals in IntermediateC-RingsA(X)via theA-Compactifications ofX

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Joshua Sack ◽  
Saleem Watson
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Completely Regular ◽  
Intermediate Ring ◽  
Maximal Ideals ◽  
Kolmogorov Theorem

LetXbe a completely regular topological space. An intermediate ring is a ringA(X)of continuous functions satisfyingC*(X)⊆A(X)⊆C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences𝒵AandℨAare defined between ideals inA(X)andz-filters onX, and it is shown that these extend the well-known correspondences studied separately forC∗(X)andC(X), respectively, to any intermediate ring. Moreover, the inverse map𝒵A←sets up a one-one correspondence between the maximal ideals ofA(X)and thez-ultrafilters onX. In this paper, we define a function𝔎Athat, in the case thatA(X)is aC-ring, describesℨAin terms of extensions of functions to realcompactifications ofX. For such rings, we show thatℨA←mapsz-filters to ideals. We also give a characterization of the maximal ideals inA(X)that generalize the Gelfand-Kolmogorov theorem fromC(X)toA(X).

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

scholarly journals Density topologies on the plane between ordinary and strong. II

2015 ◽  
Vol 62 (1) ◽  
pp. 13-25
Author(s):  
Elżbieta Wagner-Bojakowska ◽  
Władysław Wilczyński
Keyword(s):  
Topological Space ◽  
Measurable Subset ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
Completely Regular ◽  
The Family

Abstract Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.

Uniformities on a Product

1972 ◽  
Vol 24 (3) ◽  
pp. 379-389 ◽  
Author(s):  
Anthony W. Hager
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
The Real ◽  
Completely Regular

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

The Boolean Algebra of Regular Open Sets

1953 ◽  
Vol 5 ◽  
pp. 95-100 ◽  
Author(s):  
R. S. Pierce
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Distributive Lattice ◽  
Homomorphic Image ◽  
Continuous Functions ◽  
Completely Regular ◽  
Regular Open ◽  
Quasi Ordering

Let S be a completely regular topological space. Let C(S) denote the set of bounded, real-valued, continuous functions on 5. It is well known that C(S) forms a distributive lattice under the ordinary pointwise joins and meets. For any distributive lattice L and any ideal I⊆L, a quasi-ordering of L can be defined as follows : f⊇g if, for all h ∈ L, f ∩ h ∈ I implies g ∩ h ∈ I. If equivalent elements under this quasi-ordering are identified, a homomorphic image of L is obtained.

scholarly journals Quasicontinuous functions and the topology of uniform convergence on compacta

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 911-917
Author(s):  
Lubica Holá ◽  
Dusan Holý
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Continuous Functions ◽  
Hausdorff Topological Space ◽  
Cardinal Invariants ◽  
Weight Density ◽  
Quasicontinuous Functions

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.

On the Local Connectedness of βX-X

1972 ◽  
Vol 15 (4) ◽  
pp. 591-594 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Completely Regular ◽  
Local Connectedness ◽  
Hausdorff Topological Space

Let X be any completely regular Hausdorff topological space, and let βX denote its Stone-Čech compactification. This note is devoted to proving the following result:5. THEOREM. Let X be realcompact and noncompact. Then βX—X is not connected im kleinen at any point.

scholarly journals Intersections of Primary Ideals in Rings of Continuous Functions

1972 ◽  
Vol 24 (3) ◽  
pp. 502-519 ◽  
Author(s):  
R. Douglas Williams
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Necessary And Sufficient ◽  
Primary Ideals

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

scholarly journals A class of ideals in intermediate rings of continuous functions

2019 ◽  
Vol 20 (1) ◽  
pp. 109 ◽  
Author(s):  
Sagarmoy Bag ◽  
Sudip Kumar Acharyya ◽  
Dhananjoy Mandal
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Alternative Proof ◽  
Prime Ideals ◽  
Borel Sets ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Intermediate Ring ◽  
The Family ◽  
Established Fact

<p>For  any  completely  regular  Hausdorff  topological  space X,  an  intermediate  ring A(X) of  continuous  functions  stands  for  any  ring  lying between C<sup>∗</sup>(X) and C(X).  It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals.  It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X.  Interrelation between z-ideals, z◦-ideal and Ʒ<sub>A</sub>-ideals in A(X) are examined.  It is proved that within the family of almost P-spaces X, each Ʒ<sub>A</sub> -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).</p>

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