scholarly journals Usco selections of densely defined set-valued mappings

2002 ◽  
Vol 65 (2) ◽  
pp. 307-313 ◽  
Author(s):  
Warren B. Moors ◽  
Sivajah Somasundaram
Keyword(s):  
Baire Space ◽  
Topological Spaces ◽  
Range Space ◽  
Complete Space ◽  
Set Valued Mapping ◽  
Open Set ◽  
Set Valued Mappings ◽  
Densely Defined

A set-valued mapping Φ : X → 2Y acting between topological spaces X and Y is said to be “lower demicontinuous” if the interior of the closure of the set Φ−1(V): = {x ∈ X : Φ(x) ∩ V ≠ ∅} is dense in the closure of Φ−1(V) for each open set V in Y. Čoban, Kenderov and Revalski (1994) showed that for every densely defined lower demicontinuous mapping Φ acting from a Baire space X into subsets of a monotonely Čech-complete space Y, there exist a dense and Gδ subset X1 ⊆ X and an usco mapping G: X1 → 2Y such that G (x) ⊆ Φ*(x), for every x ∈ X1, where the mapping Φ*: X → 2Y is the extension of Φ defined by, W is a neighbourhood of x}.In this paper we present a proof of the above result with the notion of monotone Čcech-completeness replaced by the weaker notion of partition completeness. In addition, we observe that if the range space also lies is Stegall's class then we may assume that the mapping G is single-valued on X1.

scholarly journals A selection theorem for weak upper semi-continuous set-valued mappings

1996 ◽  
Vol 53 (2) ◽  
pp. 213-227 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Weak Topology ◽  
Weak Compactness ◽  
Baire Space ◽  
Selection Theorem ◽  
Set Valued Mapping ◽  
Set Valued Mappings

Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.

scholarly journals -spaces and the closed-graph theorem

1968 ◽  
Vol 16 (2) ◽  
pp. 89-99 ◽  
Author(s):  
S. O. Iyahen
Keyword(s):  
Topological Spaces ◽  
Topological Groups ◽  
Closed Graph ◽  
Range Space ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Complete Space ◽  
Locally Convex

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

scholarly journals On Certain Covering Properties and Minimal Sets of Bigeneralized Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1145
Author(s):  
Samer Al Ghour ◽  
Abdullah Alhorani
Keyword(s):  
Topological Spaces ◽  
Minimal Sets ◽  
Open Set ◽  
Covering Properties ◽  
Countably Compact

We introduce q-Lindelöf, u-Lindelöf, p-Lindelöf, s-Lindelöf, q-countably-compact, u-countably-compact, p-countably-compact, and s-countably-compact as new covering concepts in bigeneralized topological spaces via q-open sets and u-open sets in bigeneralized topological spaces. Relationships between them are studied. As two symmetries relationships, we show that q-Lindelöf and u-Lindelöf are equivalent concepts, and that q-countably-compact and u-countably-compact are equivalent concepts. We focus on continuity images of these covering properties. Finally, we define and investigate minimal q-open set, minimal u-open set, and minimal s-open sets as three new types of minimality in bigeneralized topological spaces.

A Characterization of Proximal Subgradient Set-Valued Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 116-122 ◽  
Author(s):  
R. A. Poliquin
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Set Valued Mapping ◽  
Selection Property ◽  
Set Valued Mappings ◽  
Bounded Below

AbstractIn this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

2015 ◽  
Vol 7 (1) ◽  
pp. 62-73 ◽  
Author(s):  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Basic Properties ◽  
New Class ◽  
Properties And Characterization ◽  
Open Set ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Characterization Theorems

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.

scholarly journals Existence of Equilibria and Fixed Points of Set-Valued Mappings on Epi-Lipschitz Sets with Weak Tangential Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Fixed Points ◽  
Tangent Cone ◽  
Compact Set ◽  
Set Valued Mapping ◽  
Clarke Tangent Cone ◽  
Set Valued Mappings ◽  
Existence Of Equilibria

We prove a new result of existence of equilibria for an u.s.c. set-valued mappingFon a compact setSofRnwhich is epi-Lipschitz and satisfies a weak tangential condition. Equivalently this provides existence of fixed points of the set-valued mappingx⇉F(x)-x. The main point of our result lies in the fact that we do not impose the usual tangential condition in terms of the Clarke tangent cone. Illustrative examples are stated showing the importance of our results and that the existence of such equilibria does not need necessarily such usual tangential condition.

scholarly journals Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

scholarly journals Fixed Point Theorems of Set-Valued Mappings in Partially Ordered Hausdorff Topological Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shujun Jiang ◽  
Zhilong Li ◽  
Shihua Luo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Monotone Mappings ◽  
Topological Spaces ◽  
Partially Ordered ◽  
Set Valued Mappings

In this work, several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings are proved in partially ordered Hausdorff topological spaces, which indeed extend and improve many recent results in the setting of metric spaces.

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