scholarly journals Covering properties of Cp(X) and Ck(X)

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3575-3599
Author(s):  
J.C. Ferrando ◽  
M. López-Pellicer
Keyword(s):  
Compact Sets ◽  
Tychonoff Space ◽  
Countable Space ◽  
Covering Properties ◽  
Second Countable Space ◽  
Bounded Sets

Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when Cp (X) or Ck (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X).

scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

scholarly journals The completion of the space of convex, bounded sets with respect to the Demyanov metric

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Jerzy Grzybowski ◽  
Andrzej Leśniewski ◽  
Tadeusz Rzeżuchowski
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Finite Dimensional ◽  
The Family ◽  
Bounded Sets ◽  
Convex Compact Sets

AbstractThe Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ

Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

1970 ◽  
Vol 22 (5) ◽  
pp. 933-938
Author(s):  
M. Edelstein
Keyword(s):  
General Setting ◽  
Locally Convex Space ◽  
Finite Family ◽  
Topological Spaces ◽  
Compact Sets ◽  
Straight Line ◽  
Straight Lines ◽  
Bounded Sets ◽  
Infinite Set ◽  
Locally Convex

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

scholarly journals More on the cardinality of a topological space

2018 ◽  
Vol 19 (2) ◽  
pp. 269 ◽  
Author(s):  
M. Bonanzinga ◽  
N. Carlson ◽  
M. V. Cuzzupè ◽  
D. Stavrova
Keyword(s):  
Hausdorff Space ◽  
Negative Answer ◽  
Topological Spaces ◽  
Compact Sets ◽  
Hausdorff Spaces ◽  
Separation Axioms ◽  
Covering Properties ◽  
Previous Inequality ◽  
The Impact ◽  
The Continuum

<p>In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”.</p><p>Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995.  The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)<sup>aL</sup><sup>c</sup><sup>(X)ψ(X)</sup>, for Hausdorff spaces and (2) |X| ≤ Hpsw(X)<sup>ω</sup><sup>L</sup><sup>c</sup><sup>(X)ψ(X)</sup>, where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2<sup>c(X)χ(X)</sup> for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2<sup>Uc(X)πχ(X)</sup> if X is a power homogeneous Urysohn space.</p>

scholarly journals Equi-Cliquishness and The Hahn Property

2016 ◽  
Vol 65 (1) ◽  
pp. 85-92
Author(s):  
Vasyl’ Nesterenko
Keyword(s):  
Metric Space ◽  
Baire Space ◽  
Residual Subset ◽  
Joint Continuity ◽  
Countable Space ◽  
Second Countable Space

Abstract We study the joint continuity of mappings of two variables. In particular, we show that for a Baire space X, a second countable space Y and a metric space Z, a map f : X × Y → Z has the Hahn property (i.e., there is a residual subset A of X such that A × Y ⊆ C(f)) if and only if f is locally equi-cliquish with respect to y and {x ∈ X: fx is continuous} is a residual subset of X.

scholarly journals On Levine′s Decomposition of Continuity

1978 ◽  
Vol 21 (4) ◽  
pp. 477-481 ◽  
Author(s):  
David Alon Rose
Keyword(s):  
Compact Space ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Compact ◽  
Weakly Continuous ◽  
Countable Space ◽  
Second Countable Space ◽  
Graph Function ◽  
Almost Continuous

AbstractA strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

Probabilistic n-normed spaces, D-n-compact sets and D-n-bounded sets

2009 ◽  
Vol 42 (5) ◽  
pp. 2729-2734 ◽  
Author(s):  
A. Pourmoslemi ◽  
M. Salimi
Keyword(s):  
Compact Sets ◽  
Normed Spaces ◽  
Bounded Sets

scholarly journals Ergodic Properties of Randomly Coloured Point Sets

2013 ◽  
Vol 65 (2) ◽  
pp. 349-402 ◽  
Author(s):  
Peter Müller ◽  
Christoph Richard
Keyword(s):  
General Framework ◽  
Point Sets ◽  
Locally Compact ◽  
Unimodular Group ◽  
Ergodic Properties ◽  
Straightforward Application ◽  
Countable Space ◽  
Local Complexity ◽  
Second Countable Space ◽  
Uniquely Ergodic

AbstractWe provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

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