Extensions of Pseudometrics

1966 ◽  
Vol 18 ◽  
pp. 981-998 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Infinite Cardinal ◽  
Product Topology

If γ is an infinite cardinal number, a subset S of a topological space X is said to be Pγ-embedded in X if every γ-separable continuous pseudometric on S can be extended to a γ-separable continuous pseudometric on X. (A pseudometric d on X is γ-separable if there exists a subset G of X such that |G| ⩽ 7 and such that G is dense in X relative to the pseudometric topology A pseudometric d is continuous if d is continuous relative to the product topology on X × X.) We say that S is P-embedded in X if every continuous pseudometric on S can be extended to a continuous pseudometric on X.

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 Number of Closed Subsets of a Topological Space

1978 ◽  
Vol 30 (02) ◽  
pp. 301-314 ◽  
Author(s):  
R. E. Hodel
Keyword(s):  
Topological Space ◽  
Basic Problem ◽  
Cardinal Number ◽  
Metrizable Space ◽  
Infinite Cardinal ◽  
Closed Sets ◽  
Complete Answer ◽  
Metrizable Spaces

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

scholarly journals On special subgroups of fundamental group

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1403-1429
Author(s):  
Zadeh Ayatollah ◽  
Fatemeh Ebrahimifar ◽  
Mohammad Mahmoodi
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Fundamental Groups ◽  
Infinite Cardinal ◽  
Homotopy Classes ◽  
Hawaiian Earring

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

Selective families of sets

1980 ◽  
Vol 372 (1750) ◽  
pp. 307-315 ◽  
Keyword(s):  
Cardinal Number ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Choice Functions ◽  
The Given

Consider a cardinal number α, a set I and a family [A v :v in I) of sets. Suppose that for every subset N of I of cardinality less than α we are given a choice of an element x f N v A v for every v in N this paper the author investigates the circumstances under which it is then always possible to make a choice of an element x*of A v for all v in which, in some precisely specified sense, can be approximated arbitrarily closely by some of the given partial choice functions x f . This question has turned out to be important when α is the least infinite cardinal number. Some of the results involve classes of ‘ large ’ cardinals.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

2020 ◽  
Vol 27 (03) ◽  
pp. 495-508
Author(s):  
Ahmed Maatallah ◽  
Ali Benhissi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cardinal Number ◽  
Prime Ideals ◽  
Infinite Cardinal ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Power Series Rings ◽  
Formal Power ◽  
Infinite Set

Let A be a ring. In this paper we generalize some results introduced by Aliabad and Mohamadian. We give a relation between the z-ideals of A and those of the formal power series rings in an infinite set of indeterminates over A. Consider A[[XΛ]]3 and its subrings A[[XΛ]]1, A[[XΛ]]2, and A[[XΛ]]α, where α is an infinite cardinal number. In fact, a z-ideal of the rings defined above is of the form I + (XΛ)i, where i = 1, 2, 3 or an infinite cardinal number and I is a z-ideal of A. In addition, we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients. As a natural result, we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

scholarly journals A congruence-free semigroup associated with an infinite cardinal number

1983 ◽  
Vol 93 (3-4) ◽  
pp. 245-257 ◽  
Author(s):  
M. Paula O. Marques
Keyword(s):  
Homomorphic Image ◽  
Free Semigroup ◽  
Cardinal Number ◽  
Infinite Cardinal ◽  
Rees Factor ◽  
Infinite Cardinality ◽  
The Ideal

SynopsisLet X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠

scholarly journals Free paratopological groups

2015 ◽  
Vol 16 (2) ◽  
pp. 89
Author(s):  
Ali Sayed Elfard
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Inductive Limit ◽  
Infinite Cardinal ◽  
Alexandroff Space ◽  
Limit Property ◽  
Neighborhood Base ◽  
Paratopological Group

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces $X$ for which the group FP(X) has the inductive limit property.

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.

On partitioning the infinite subsets of large cardinals

1984 ◽  
Vol 49 (2) ◽  
pp. 539-541 ◽  
Author(s):  
R. J. Watro
Keyword(s):  
Order Type ◽  
Large Cardinal ◽  
Infinite Cardinal ◽  
Product Topology ◽  
Weakly Compact ◽  
Ramsey Theorem ◽  
Borel Sets ◽  
Analytic Sets ◽  
The Axiom Of Choice ◽  
Usual Product

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.

Inner Models and Large Cardinals

1995 ◽  
Vol 1 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Ronald Jensen
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Infinite Cardinal ◽  
Recursive Definition ◽  
Von Neumann ◽  
Ordinal Numbers ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

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