One point compactification for generalized quotient spaces (original) (raw)
Generalized quotient topologies
Acta Mathematica Hungarica, 2010
A definition of a generalized quotient topology is given and some characterizations of this concept, up to generalized homeomorphisms, are furnished. For the first approach, we exhibit a monotonic map spanning that generalized quotient topology. We also prove that the notions of generalized normality and generalized compactness are preserved by those quotient structures.
Complete metrizability of generalized compact-open topology
Topology and its Applications, 1999
Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y: a partial map is a pair (B,f). where B E CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to El'. Denote by 7~ the generalized compact-open topology on P. We show that if X is a hemicompact metrizable space and Y is a FrCchet space. then (P. TC) is completely metrizable and homeomorphic to a closed subspace of (CL(X), TF) x (C(X. Y). T~,cJ), where T,T is the Felt topology on CL(X) and 71'0 is the compact-open topology on C(X, Y).
Fundamentals of Contemporary Mathematical Sciences
In a recent paper, a novel class of generalized compact sets (briefly, g-Tg -compact sets) in generalized topological spaces (briefly, Tg -spaces) has been studied. In this paper, the concept is further studied and, other derived concepts called countable, sequential, and local generalized compactness (countable, sequential, local g-Tg -compactness) in Tg -spaces are also studied relatively. The study reveals that g-Tg -compactness implies local g-Tg -compactness and countable g-Tg-compactness, sequential g-Tg -compactness implies countable g-Tg -compactness and, g-Tg -compactness is a generalized topological property (briefly, Tg -property). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in the literature, and a nice application supports the overall theory.
The compactificability classes of certain spaces
International Journal of Mathematics and Mathematical Sciences, 2006
We apply the theory of the mutual compactificability to some spaces, mostly derived from the real line. For example, any noncompact locally connected metrizable generalized continuum, the Tichonov cube without its zero point I ℵ0 \{0}, as well as the Cantor discontinuum without its zero point D ℵ0 \{0} are of the same class of mutual compactificability as R.
Topology and its Applications, 2008
A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.
Compactness and compactifications in generalized topology
Topology and its Applications, 2015
A generalized topology in a set X is a collection Cov X of families of subsets of X such that the triple (X, Cov X , Cov X ) is a generalized topological space (a gts) in the sense of Delfs and Knebusch. In this work, a new notion of a strict compactification of a generalized topological space is introduced and investigated in ZF. The Ultrafilter Theorem (in abbreviation UFT) is shown to be equivalent to the compactness of every Wallman extension of an arbitrary semi-normal space. That every small weakly normal gts has its Wallman strict compactification as well as several other sentences are proved to be equivalent to UFT. Among results concerning categories, it is shown that the construct of partially topological gtses is topological in ZF. Many illuminating examples are given and open problems are posed. 2010 Mathematics Subject Classification: Primary 54D35, 03E25; Secondary 54A05, 03E30.
Compact covers and function spaces
Journal of Mathematical Analysis and Applications, 2014
For a Tychonoff space X, we denote by C p (X) and C c (X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of C p (X), we extend Okunev's results by showing that if there exists a surjection from C p (X) onto C p (Y) (resp. from L p (X) onto L p (Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c (X) onto C c (Y), then Y is a separable completely metrizable space. We study also the non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by p-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and p-equivalent to a first countable Y , then Y is metrizable (V. Valov). Some additional results are presented.
Čech-Completeness and Related Properties of the Generalized Compact-Open Topology
Journal of Applied Analysis, 2010
The generalized compact-open topology τ C on partial continuous functions with closed domains in X and values in Y is studied. If Y is a noncountably compactČech-complete space with a G δ-diagonal, then τ C isČechcomplete, sieve complete and satisfies the p-space property of Arhangel'skiǐ, respectively, if and only if X is Lindelöf and locally compact. Lindelöfness, paracompactness and normality of τ C is also investigated. New results are obtained onČech-completeness, sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y .
A generalization of -metrizable spaces
2019
We introduce a new class of -metrizable spaces, namely countably -metrizable spaces. We show that the class of all -metrizable spaces is a proper subclass of counably -metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with -metrizable spaces. We prove a generalization of a Chigogidze result that the Čech-Stone compactification of a pseudocompact countably -metrizable space is -metrizable.