Convergence and divergence sets of sequences of real continuous functions on a metric space

1975 ◽  
Vol 17 (2) ◽  
pp. 120-126
Author(s):  
M. A. Lunina
2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG

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.


2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.


1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).


1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.


2017 ◽  
Vol 28 (3) ◽  
pp. 340-383 ◽  
Author(s):  
PABLO ARRIGHI ◽  
SIMON MARTIEL ◽  
VINCENT NESME

It is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions over a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that they have a well-defined notion of translation. We propose a notion of graphs, which preserves or generalizes these features. Whereas Cayley graphs are very regular, generalized Cayley graphs are arbitrary, although of a bounded degree. We extend cellular automata theory to these arbitrary, bounded degree, time-varying graphs. The obtained notion of cellular automata is stable under composition and under inversion.


1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.


2021 ◽  
pp. 3031-3038
Author(s):  
Raghad I. Sabri

      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.


2009 ◽  
Vol 44 (1) ◽  
pp. 159-168 ◽  
Author(s):  
Miloslav Duchoň ◽  
Peter Maličký

Abstract We present a Helly type theorem for sequences of functions with values in metric spaces and apply it to representations of some mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. More concretely, a representation of certain majored linear operators on the space of continuous functions into a complete metric space.


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