More on Convergence of Continuous Functions and Topological Convergence of Sets

1985 ◽  
Vol 28 (1) ◽  
pp. 52-59 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Baire Category ◽  
Continuous Functions ◽  
Locally Compact ◽  
Topological Convergence ◽  
Convergence Of Sets ◽  
Compact Subsets

AbstractLet C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.

On Uniform Convergence of Continuous Functions and Topological Convergence of Sets

1983 ◽  
Vol 26 (4) ◽  
pp. 418-424 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Related Result ◽  
Continuous Functions ◽  
Topological Convergence ◽  
The Relationship ◽  
Convergence Of Sets

AbstractLet X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.

scholarly journals Compactness Property of Fuzzy Soft Metric Space and Fuzzy Soft Continuous Function

2021 ◽  
pp. 3031-3038
Author(s):  
Raghad I. Sabri
Keyword(s):  
Functional Analysis ◽  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Property ◽  
Locally Compact ◽  
Fuzzy Metric Spaces ◽  
Fundamental Properties ◽  
Definition Of

      The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics.    Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.

Local Compactness in Set Valued Function Spaces

1976 ◽  
Vol 19 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Saroop K. Kaul
Keyword(s):  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Range Space ◽  
Locally Compact ◽  
Local Compactness

Recently Hunsaker and Naimpally [2] have proved: The pointwise closure of an equicontinuous family of point compact relations from a compact T2-space to a locally compact uniform space is locally compact in the topology of uniform convergence. This is a generalization of the same result of Fuller [1] for single valued continuous functions.For a range space which is locally compact normal and uniform theorem B below is an improvement on the result of Hunsaker and Naimpally quoted above [see Remark 3 at the end of this paper].

scholarly journals Dynamics of induced systems

2016 ◽  
Vol 37 (7) ◽  
pp. 2034-2059 ◽  
Author(s):  
ETHAN AKIN ◽  
JOSEPH AUSLANDER ◽  
ANIMA NAGAR
Keyword(s):  
Phase Space ◽  
Uniform Convergence ◽  
Metric Space ◽  
Cantor Set ◽  
Main Theme ◽  
Hausdorff Topology ◽  
Continuous Maps ◽  
Dynamical Properties ◽  
Compact Subsets

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if$X$is a metric space, let$2^{X}$denote the space of non-empty compact subsets of$X$provided with the Hausdorff topology. If$f$is a continuous self-map on$X$, there is a naturally induced continuous self-map$f_{\ast }$on$2^{X}$. Our main theme is the interrelation between the dynamics of$f$and$f_{\ast }$. For such a study, it is useful to consider the space${\mathcal{C}}(K,X)$of continuous maps from a Cantor set$K$to$X$provided with the topology of uniform convergence, and$f_{\ast }$induced on${\mathcal{C}}(K,X)$by composition of maps. We mainly study the properties of transitive points of the induced system$(2^{X},f_{\ast })$both topologically and dynamically, and give some examples. We also look into some more properties of the system$(2^{X},f_{\ast })$.

On a question of R. Godement about the spectrum of positive, positive definite functions

1991 ◽  
Vol 110 (1) ◽  
pp. 137-142
Author(s):  
Mohammed B. Bekka
Keyword(s):  
Compact Group ◽  
Uniform Convergence ◽  
Unitary Representation ◽  
Convex Set ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Image Position ◽  
Compact Subsets

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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 On Continuous Isomorphisms of Topological Groups

1950 ◽  
Vol 1 ◽  
pp. 109-111
Author(s):  
Morikuni Gotô ◽  
Hidehiko Yamabe
Keyword(s):  
Uniform Convergence ◽  
Topological Groups ◽  
Natural Topology ◽  
Locally Compact ◽  
Connected Group ◽  
Inner Automorphisms

Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

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