The Open Mapping and Closed Graph Theorem for Embeddable Topological Semigroups

1975 ◽  
Vol 18 (5) ◽  
pp. 671-674
Author(s):  
Douglass L. Grant
Keyword(s):  
Independent Interest ◽  
Open Mapping ◽  
Topological Groups ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Preliminary Results ◽  
Topological Semigroups

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.

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.

Ultrabarrelled groups and the closed graph theorem

1969 ◽  
Vol 65 (1) ◽  
pp. 53-58 ◽  
Author(s):  
S. O. Iyahen
Keyword(s):  
Topological Group ◽  
Group Homomorphism ◽  
Topological Groups ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.

B(?)-spaces and the closed graph theorem

1964 ◽  
Vol 153 (4) ◽  
pp. 293-298 ◽  
Author(s):  
Taqdir Husain
Keyword(s):  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem

scholarly journals The closed graph theorem without category

1987 ◽  
Vol 36 (2) ◽  
pp. 283-287 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Baire Category ◽  
Baire Category Theorem ◽  
Normed Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Category Theorem

We show that a diagonal theorem of P. Antosik can be used to give a proof of the Closed Graph Theorem for normed spaces which does not depend upon the Baire Category Theorem.

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