Stetige Konvergenz und der Satz von Ascoli und Arzelà

Poppe, Harry

doi:10.1002/mana.19650300107

Cited by 13 publications

(2 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [14], Kelley defined even continuous mappings between topological spaces, and Poppe [19] generalized this notion to sets of continuous mappings between limit spaces. The following is straightforward transcription of these definitions to L-spaces.…”

Section: Even Continuitymentioning

confidence: 99%

An Ascoli theorem for sequential spaces

Sonck

2001

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

Abstract. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C * -algebras.2000 Mathematics Subject Classification. 54A20, 54C35, 54E15.1. Introduction. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. Such general theorems are inspired by the classical Ascoli theorem, proved by G. Ascoli (and independently by C. Arzelà) in the 19th century (see [3,4]). It characterizes compactness of sets of continuous real-valued functions on the interval [0, 1] with respect to the topology of uniform convergence. Since then, many related theorems have been proved, for example, characterizing compactness of sets of continuous functions from a topological to a uniform space (see [6]), of uniformly continuous functions from a merotopic to a uniform space (see [5]) of continuous functions between topological spaces (see [14,17]). As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natural function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct. Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categorical setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory. Wyler himself developed new examples of Ascoli theorems for sets of continuous functions between limit spaces and of uniformly continuous functions from a uniform convergence space to a pseudo-uniform space (see [24]).In this paper, we present another setting for an Ascoli theorem: we choose the construct L of sequential spaces. Sequential spaces were already introduced at the beginning of the century by Fréchet (see [9,10]) and Urysohn (see [2,23]), even before topological spaces were axiomatized. Since then they have extensively been used as a tool in topology and analysis. Since the 60s sequential structures have been investigated from a categorical point of view (for categorical background, we refer to [1]);

show abstract

Section: Even Continuitymentioning

confidence: 99%

An Ascoli theorem for sequential spaces

Sonck

2001

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

show abstract

“…Since G and (G, ¿) have the same compact sets, the topology on G^ is just its relative topology as a subset of C((G, k), T). Thus, by an Ascoli Theorem of sufficient generality (see for instance [3], [14] or [12]), and the fact that every subset of C((G, k), T) is bounded, a subset of G^ is compact if and only if it is closed in C((G, k), T) and evenly continuous on (G, ¿). By the lemma just proved, each set in 3$~ is evenly continuous on G and hence on (G, ¿).…”

Section: Conventionmentioning

confidence: 99%

$k$-groups and duality

Noble¹

1970

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Recall that a function is Ac-continuous if its restriction to each compact subset of its domain is continuous. We call a topological group G a Ac-group if each ¿-continuous homomorphism on G is continuous. As we will see in §1, where elementary properties of Ac-groups are studied, Ac-groups are the appropriate topological group analogue to Ac-spaces. As one would expect, they provide a useful tool for the study of dual groups topologized with the compact-open topology. In §2 we show that the duality map on an abelian Ac-group is always continuous, and in §3 we give some extensions of the Pontrjagin-Van Kampen duality theorem. As a corollary we show that each closed subgroup of a countable product of locally compact abelian groups satisfies duality, extending the principal result of [8]-that the inverse limit of a sequence of locally compact abelian groups satisfies duality.1. Elementary properties. Recall that where (A", t) is a topological space and k(t) = {U^ X : For each compact Fs X, U n K is relatively open in K}, (X, k(t)) is a topological space and k(t) is called the Ac-extension of t. (Also, when (A", t) is denoted as X, (X, k(t)) is denoted as AcA" and is called the Ac-extension of A".) The topology k(t) is the largest topology on X coinciding on compact sets with t and X is called a Ac-space if A"= AcA". Obviously A" is a ¿-space if and only if each Ac-continuous function on X is continuous. Where (G, t) is a topoligical group, let kg(t) denote the supremum of all group topologies t' satisfying t' -¿k(t); and where (G, t) is denoted as G, denote (G, kg(t)) as kgG. Proposition.If (G, t) is any topological group, then kg(t) is the largest group topology for G coinciding on compact sets with t. Thus (G, t) is a k-group if and only ifkg(t) = t.Proof. Since the supremum of any collection of group topologies is a group topology, kg(t) is a group topology ; since t S kg(t) ^ k(t), kg(t) coincides on compact sets with / and by definition kg(t) is the largest group topology with this property. If (G, t) is a Ac-group, then the identity map from (C7, t) to (G, kg(t)), which is a Ac-continuous homomorphism, is continuous, so t=kg(t). Conversely, suppose t=kg(t) and let/be a ¿-continuous homomorphism on G. Let t' be the smallest

show abstract

Supercategories of Top and the Inevitable Emergence of Topological Constructs

Lowen-Colebunders

Löwen

2001

Handbook of the History of General Topology

View full text Add to dashboard Cite

Stetige Konvergenz und der Satz von Ascoli und Arzelà

Cited by 13 publications

References 15 publications

An Ascoli theorem for sequential spaces

An Ascoli theorem for sequential spaces

$k$-groups and duality

Supercategories of Top and the Inevitable Emergence of Topological Constructs

Contact Info

Product

Resources

About