Function space topologies

An ascoli theorem for multi-valued functions

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010351 ◽

1971 ◽

Vol 12 (4) ◽

pp. 466-472 ◽

Cited By ~ 5

Author(s):

Vincent J. Mancuso

Keyword(s):

Function Space ◽

Metric Spaces ◽

Type Theorem ◽

Function Space Topologies

The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind:

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A study of function space topologies for multifunctions

Applied General Topology ◽

10.4995/agt.2017.7149 ◽

2017 ◽

Vol 18 (2) ◽

pp. 331 ◽

Cited By ~ 1

Author(s):

Ankit Gupta ◽

Ratna Dev Sarma

Keyword(s):

Function Space ◽

Title Page ◽

Compact Open Topology ◽

Continuous Convergence ◽

Function Space Topologies

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (</span><span>τ,μ</span><span>)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. </span></p></div></div></div>

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Function Space Topologies for Connectivity and Semi-Connectivity Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1966-044-4 ◽

1966 ◽

Vol 9 (3) ◽

pp. 349-352 ◽

Cited By ~ 2

Author(s):

Somashekhar Amrith Naimpally

Keyword(s):

Uniform Convergence ◽

Function Space ◽

Uniform Space ◽

Continuous Functions ◽

Topological Spaces ◽

Subsequent Paper ◽

Graph Topology ◽

Function Space Topologies

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

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