scholarly journals Perfect Mappings and Spaces of Countable Type

1970 ◽  
Vol 22 (6) ◽  
pp. 1208-1210 ◽  
Author(s):  
J. E. Vaughan
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
Regular Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Countable Type ◽  
Open Set

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

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.

Epimorphisms From S(X) onto S(Y)

1986 ◽  
Vol 38 (3) ◽  
pp. 538-551 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Severe Restriction ◽  
Completely Regular ◽  
Image Position

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for whichforms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.

Descriptive sets

1966 ◽  
Vol 291 (1426) ◽  
pp. 353-367 ◽  
Keyword(s):  
Metric Space ◽  
Regular Space ◽  
Countable Union ◽  
Borel Set ◽  
Borel Sets ◽  
Countable Intersection ◽  
Completely Regular ◽  
Closed Sets ◽  
Open Set

A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.

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.

Functional characterizations of p-spaces

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Ľubica Holá
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Open Subset ◽  
Continuous Extension ◽  
Regular Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Complete Space ◽  
Completely Regular ◽  
Locally Compact Space

AbstractWe show that a completely regular space Y is a p-space (a Čech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.

scholarly journals Quasicompactness and functionally Hausdorff spaces

1973 ◽  
Vol 15 (3) ◽  
pp. 319-324 ◽  
Author(s):  
A. J. D'Aristotle
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Separation Axioms

In [1] Van Est and Freudenthal introduced and studied several new separation axioms for a topological space. One of these was the pτsq axiom: Given distinct points p and q of X, there exists a real continuous function f on X with f(p) ≠ f(q). They observed that the pτsq axiom lies strictly between the Hausdorff and completely regular axioms, and that it neither implies nor is implied by the T3 axiom.

A characterization of completely regular spaces

2015 ◽  
Vol 26 (03) ◽  
pp. 1550032 ◽  
Author(s):  
Richard W. M. Alves ◽  
Victor H. L. Rocha ◽  
Josiney A. Souza
Keyword(s):  
Topological Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Admissible Family

This paper proves that uniform spaces and admissible spaces form the same class of topological spaces. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and dynamics in completely regular spaces.

scholarly journals BOUNDING THE MINIMUM ORDER OF Aπ-BASE

2011 ◽  
Vol 83 (2) ◽  
pp. 321-328
Author(s):  
MARÍA MUÑOZ
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Minimum Order ◽  
Compact Spaces ◽  
Cardinal Invariants ◽  
Open Set ◽  
Open Questions ◽  
The Family ◽  
The One

AbstractLetXbe a topological space. A family ℬ of nonempty open sets inXis called aπ-base ofXif for each open setUinXthere existsB∈ℬ such thatB⊂U. The order of aπ-base ℬ at a pointxis the cardinality of the family ℬx={B∈ℬ:x∈B} and the order of theπ-base ℬ is the supremum of the orders of ℬ at each pointx∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in:Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’,Amer. Math. Soc. Transl.134(1987), 93–118] establishes that the minimum order of aπ-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projectiveπ-character bounds the order of aπ-base’,Proc. Amer. Math. Soc.136(2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of aπ-base is bounded by the ‘projectiveπ-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projectiveπ-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projectiveπ-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of aπ-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.

Continuous images of proper analytic and proper Borel spaces

1974 ◽  
Vol 75 (2) ◽  
pp. 185-191 ◽  
Author(s):  
J. E. Jayne
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Continuous Images

All topological spaces considered will be completely regular Hausdorff spaces. The word space will be used to refer to such a topological space.

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.

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