scholarly journals Solid convergence spaces

1973 ◽  
Vol 8 (3) ◽  
pp. 443-459 ◽  
Author(s):  
M. Schroder
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Convergence Spaces

The category of solid convergence spaces is introduced, and shown to lie strictly between the category of all convergence spaces and that of pseudo-topological spaces. A wide class of convergence spaces, including the c-embedded spaces of Binz, is then characterized in terms of this concept. Finally, several illustrative examples are given.

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

2003 ◽  
Vol 10 (2) ◽  
pp. 209-222
Author(s):  
I. Bakhia
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

scholarly journals Homological epimorphisms, homotopy epimorphisms and acyclic maps

2020 ◽  
Vol 32 (6) ◽  
pp. 1395-1406
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Loop Spaces ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Algebraic Description ◽  
Acyclic Map ◽  
Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

scholarly journals Convergence Classes of L -Filters in L , M -Fuzzy Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Yang ◽  
Sheng-Gang Li ◽  
William Zhu ◽  
Xiao-Fei Yang ◽  
Ahmed Mostafa Khalil
Keyword(s):  
Topological Spaces ◽  
Convergence Spaces ◽  
Topological Convergence ◽  
Convergence Structure ◽  
Fuzzy Topological Spaces ◽  
The Relationship

An L , M -fuzzy topological convergence structure on a set X is a mapping which defines a degree in M for any L -filter (of crisp degree) on X to be convergent to a molecule in L X . By means of L , M -fuzzy topological neighborhood operators, we show that the category of L , M -fuzzy topological convergence spaces is isomorphic to the category of L , M -fuzzy topological spaces. Moreover, two characterizations of L -topological spaces are presented and the relationship with other convergence spaces is concretely constructed.

scholarly journals A better framework for first countable spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 289
Author(s):  
Gerhard Preuss
Keyword(s):  
Topological Space ◽  
Uniform Space ◽  
Topological Spaces ◽  
Convergence Space ◽  
Convergence Spaces ◽  
Natural Function ◽  
Topological Construct ◽  
Countable Space ◽  
Filter Space ◽  
First Countable Space

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

scholarly journals Two preservation results for countable products of sequential spaces

2007 ◽  
Vol 17 (1) ◽  
pp. 161-172 ◽  
Author(s):  
MATTHIAS SCHRÖDER ◽  
ALEX SIMPSON
Keyword(s):  
Topological Spaces ◽  
Product Spaces ◽  
Convergence Spaces ◽  
Countable Product ◽  
Discrete Spaces ◽  
Sequential Spaces

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.

Simplicity of Categories Defined by Symmetry Axioms

1991 ◽  
Vol 34 (2) ◽  
pp. 240-248
Author(s):  
E. Lowen-Colebunders ◽  
Z. G. Szabo
Keyword(s):  
Topological Spaces ◽  
Convergence Spaces ◽  
Epireflective Subcategory ◽  
Closure Spaces ◽  
Symmetry Axiom

AbstractWe consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.

scholarly journals Perfect maps on convergence spaces

1979 ◽  
Vol 20 (3) ◽  
pp. 447-466
Author(s):  
Robert A. Herrmann
Keyword(s):  
Final Section ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Convergence Space ◽  
Covering Property ◽  
Convergence Spaces ◽  
Weakly Continuous ◽  
Perfect Map ◽  
Perfect Maps ◽  
Compact Map

The concept of the perfect map on a convergence space (X, q), where q is a convergence function, is introduced and investigated. Such maps are not assumed to be either continuous or surjective. Some nontrivial examples of well known mappings between topological spaces, nontopological pretopological spaces and nonpseudotopological convergence spaces are shown to be perfect in this new sense. Among the numerous results obtained is a covering property for perfectness and the result that such maps are closed, compact, and for surjections almost-compact. Sufficient conditions are given for a compact (respectively almost-compact) map to be perfect. In the final section, a major result shows that if f: (X, q) → (Y, p) is perfect and g: (X, q) → (Z, s) is weakly-continuous into Hausdorff Z, then (f, g): (X, q) → (Y×Z, p×s) is perfect. This result is given numerous applications.

scholarly journals One-point compactification on convergence spaces

1994 ◽  
Vol 17 (2) ◽  
pp. 277-282
Author(s):  
Shing S. So
Keyword(s):  
Topological Spaces ◽  
Convergence Space ◽  
Locally Compact ◽  
Convergence Spaces ◽  
The One ◽  
One Point Compactification

A convergence space is a set together with a notion of convergence of nets. It is well known how the one-point compactification can be constructed on noncompact, locally compact topological spaces. In this paper, we discuss the construction of the one-point compactification on noncompact convergence spaces and some of the properties of the one-point compactification of convergence spaces are also discussed.

Borsuk–Ulam theorem for filtered spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos Biasi ◽  
Alice Kimie Miwa Libardi ◽  
Denise de Mattos ◽  
Sergio Tsuyoshi Ura
Keyword(s):  
Wide Class ◽  
Hausdorff Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Equivariant Map ◽  
Ulam Theorem ◽  
Free Involution

Abstract Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 )   for  ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

Accessible sets and (Lω1ω)t-equivalence for T3 spaces

1984 ◽  
Vol 49 (3) ◽  
pp. 961-967 ◽  
Author(s):  
Juan Carlos Martínez
Keyword(s):  
Natural Number ◽  
Finite Type ◽  
Wide Class ◽  
Expressive Power ◽  
Topological Spaces

Ziegler studies in [2] the expressive power of (Lωω)t for T3 topological spaces. He defines for every natural number n the set of n-types by induction: . If A is a T3 space, the n-type of a ∈ A is defined inductively by: : in every neighborhood of a there is an a′ ≠ a with tn(a′, A) = α}. These types are (Lωω)t-definable. Then, it is shown that two T3 spaces are (Lωω)t-equivalent precisely if for every n-type α they have the same number of points with n-type α (cf. [2]).In order to study the expressability of (Lω1ω)t, for T3 spaces, we introduce in this paper the notion of accessible set. Looking at the behaviour of convergence by means of this notion, we refine Ziegler's notion of n-type and introduce a new set Sn of n-types, which are (Lω1ω)t-definable. Then, we prove a characterization of (Lω1ω)t-equivalence for a wide class of T3 spaces. A T3 space A belongs to this class if there is a κ ∈ ω such that, for every n ∈ ω, there are at most κ n-types in Sn which are satisfiable in A. Such a space is said to be of a-finite type. Some relations between these spaces and the spaces of finite type in the sense of [2] are shown in the last section.The contents of the present paper are treated in more detail in [3].

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