Diamond Principles, Ideals and the Normal Moore Space Problem

1981 ◽  
Vol 33 (2) ◽  
pp. 282-296 ◽  
Author(s):  
Alan D. Taylor
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
Space Problem ◽  
Moore Space ◽  
Image Position

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such thatThe sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.

A Collectionwise Hausdorff, Non-Normal Moore Space

1976 ◽  
Vol 28 (3) ◽  
pp. 632-634 ◽  
Author(s):  
Michael L. Wage
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
Moore Space ◽  
Collectionwise Hausdorff

A topological space X is said to be collectionwise Hausdorff if every discrete collection of points of X can be simultaneously separated by a collection of pairwise disjoint open sets. The question of whether there exists a collectionwise Hausdorff, non-normal Moore space was first asked by R. L. Moore. In 1964, J. M. Worrell announced that such a space did indeed exist (see [7]), but his proof has never appeared in print.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

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.

scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

scholarly journals Relations on topological spaces: Urysohn's lemma

1968 ◽  
Vol 8 (1) ◽  
pp. 37-42
Author(s):  
Y.-F. Lin
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Order Relation ◽  
Topological Spaces ◽  
Topological Closure ◽  
Image Position

Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

A Provisional Solution to the Normal Moore Space Problem

2002 ◽  
Vol 8 (3) ◽  
pp. 443
Author(s):  
Gary Gruenhage ◽  
Peter J. Nyikos ◽  
William G. Fleissner ◽  
Alan Dow ◽  
Franklin D. Tall ◽  
...  
Keyword(s):  
Space Problem ◽  
Moore Space

scholarly journals A generalization of a theorem of Aquaro

1973 ◽  
Vol 9 (1) ◽  
pp. 105-108 ◽  
Author(s):  
C.E. Aull
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuum Hypothesis ◽  
Open Cover ◽  
Moore Space ◽  
Countably Compact Space ◽  
Classic Result ◽  
Countably Compact ◽  
The Continuum

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

scholarly journals Relative bifurcation sets and the local dimension of univoque bases

2020 ◽  
pp. 1-33
Author(s):  
PIETER ALLAART ◽  
DERONG KONG
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Pairwise Disjoint ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Local Dimension ◽  
Dimension Zero ◽  
Image Position ◽  
Dimension One

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

Topological H-Surfaces

1965 ◽  
Vol 17 ◽  
pp. 847-849 ◽  
Author(s):  
H. G. Helfenstein
Keyword(s):  
Topological Space ◽  
Binary Composition ◽  
Image Position

An H-space is a topological space T for which it is possible to define a continuous binary compositionwith the following properties: there exists a homotopy unit, i.e. an element u ∊ T such that

Sign in / Sign up

Export Citation Format

Share Document