On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Süleyman Önal ◽  
Çetin Vural
Keyword(s):  
Topological Space ◽  
Countable Family ◽  
Product Spaces ◽  
Finitely Generated ◽  
Countable Product ◽  
Covering Properties

AbstractWe introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.

Real Functions, Covers and Bornologies

2021 ◽  
Vol 78 (1) ◽  
pp. 199-214
Author(s):  
Lev Bukovský
Keyword(s):  
Topological Space ◽  
Pointwise Convergence ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Functions ◽  
Covering Properties ◽  
Real Functions ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

scholarly journals Topological Games on Product Spaces and its Fuzzification

2013 ◽  
Vol 2 ◽  
pp. 11-15
Author(s):  
Bidyanand Prasad ◽  
BP Kumar
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Perfect Information ◽  
Topological Spaces ◽  
Product Spaces ◽  
Pursuit And Evasion

This paper is concerned with the introduction of an infinite positional game of pursuit and evasion over an ideal of a topological space. A topological game has been played over some new D-product and C-product spaces of two Hausdorff topological spaces. Perfect information, decisions and goals in a game may not be feasible. Hence, fuzzy set theory has been applied in this paper to obtain better results. Academic Voices, Vol. 2, No. 1, 2012, Pages 11-15 DOI: http://dx.doi.org/10.3126/av.v2i1.8278

Dense Subspaces of Product Spaces

1983 ◽  
Vol 35 (6) ◽  
pp. 986-1000 ◽  
Author(s):  
Toshiji Terada
Keyword(s):  
The Other ◽  
Countable Family ◽  
Remarkable Difference ◽  
Product Spaces ◽  
Other Hand ◽  
Metrizable Spaces

Unless otherwise specified, all spaces considered here are regular T1-spaces. A space X is called σ-discrete if X is the union of a countable family of discrete subspaces. Arhangel'skii [2] showed that the class of spaces which contain dense σ-discrete subspaces is productive. The fact that the class of spaces which contain dense subspaces of countable pseudocharacter is productive is obtained by Amirdzanov [1]. On the other hand, the class of spaces which contain metrizable spaces as dense subspaces is obviously not productive. As a generalized concept of metrizable spaces there is the concept of σ-spaces [14]. This class of spaces has many similar properties to the class of metrizable spaces. However we will point out a remarkable difference between the class of metrizable spaces and the class of σ-spaces by showing that the class of spaces which contain σ-spaces as dense subspaces is productive.

scholarly journals Selective covering properties of product spaces

2014 ◽  
Vol 165 (5) ◽  
pp. 1034-1057 ◽  
Author(s):  
Arnold W. Miller ◽  
Boaz Tsaban ◽  
Lyubomyr Zdomskyy
Keyword(s):  
Product Spaces ◽  
Covering Properties

scholarly journals Measures on countable product spaces

1969 ◽  
Vol 30 (3) ◽  
pp. 639-644 ◽  
Author(s):  
Edwin Elliott
Keyword(s):  
Product Spaces ◽  
Countable Product

scholarly journals A continuity property related to an index of non-separability and its applications

1992 ◽  
Vol 46 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Function ◽  
Compact Subset ◽  
Metric Space ◽  
Dense Subset ◽  
Countable Family ◽  
Set Valued Mapping ◽  
Continuous Convex Function ◽  
Fréchet Differentiable

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

scholarly journals On Fuzzy L-paracompact Topological Spaces

2021 ◽  
Vol 8 ◽  
pp. 38-40
Author(s):  
Francisco Gallego Lupiáñez
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Locally Compact ◽  
Hausdorff Topological Space ◽  
Characteristic Map ◽  
Covering Properties ◽  
Fuzzy Topological Space

The aim of this paper is to study fuzzy extensions of some covering properties defined by L. Kalantan as a modification of some kinds of paracompactness-type properties due to A.V.Arhangels'skii and studied later by other authors. In fact, we obtain that: if (X,T) is a topological space and A is a subset of X, then A is Lindelöf in (X,T) if and only if its characteristic map χ_{A} is a Lindelöf subset in (X,ω(T)). If (X,τ) is a fuzzy topological space, then, (X,τ) is fuzzy Lparacompact if and only if (X,ι(τ)) is L-paracompact, i.e. fuzzy L-paracompactness is a good extension of L-paracompactness. Fuzzy L₂-paracompactness is a good extension of L₂- paracompactness. Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava' or in the Wagner and McLean' sense) which is fuzzy locally compact (in the Kudri and Wagner' sense) is fuzzy L₂-paracompact

scholarly journals A note on the countable union of prime submodules

2001 ◽  
Vol 27 (10) ◽  
pp. 641-643 ◽  
Author(s):  
M. R. Pournaki ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Countable Family ◽  
Prime Ideals ◽  
Countable Union ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Prime Submodules

LetMbe a finitely-generated module over a Noetherian ringR. Suppose𝔞is an ideal ofRand letN=𝔞Mand𝔟=Ann(M/N). If𝔟⫅J(R),Mis complete with respect to the𝔟-adic topology,{Pi}i≥1is a countable family of prime submodules ofM, andx∈M, thenx+N⫅∪i≥1Piimplies thatx+N⫅Pjfor somei≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

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.

Even Covering Properties and Somewhat Normal Spaces

1986 ◽  
Vol 29 (2) ◽  
pp. 154-159
Author(s):  
Hans-Peter Künzi ◽  
Peter Fletcher
Keyword(s):  
Topological Space ◽  
Open Cover ◽  
Normal Space ◽  
Completely Regular ◽  
Covering Properties ◽  
Normal Cover

AbstractA topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.

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