scholarly journals θ-regular spaces

1985 ◽  
Vol 8 (3) ◽  
pp. 615-619 ◽  
Author(s):  
Dragan S. Janković
Keyword(s):  
Topological Space ◽  
Weak Continuity ◽  
Topological Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Weakly Continuous ◽  
Graph Function

In this paper we define a topological spaceXto beθ-regular if every filterbase inXwith a nonemptyθ-adherence has a nonempty adherence. It is shown that the class ofθ-regular topological spaces includes rim-compact topological spaces and thatθ-regularH(i)(Hausdorff) topological spaces are compact (regular). The concept ofθ-regularity is used to extend a closed graph theorem of Rose [1]. It is established that anr-subcontinuous closed graph function into aθ-regular topological space is continuous. Another sufficient condition for continuity of functions due to Rose [1] is also extended by introducing the concept of almost weak continuity which is weaker than both weak continuity of Levine and almost continuity of Husain. It is shown that an almost weakly continuous closed graph function into a strongly locally compact topological space is continuous.

scholarly journals On Levine′s Decomposition of Continuity

1978 ◽  
Vol 21 (4) ◽  
pp. 477-481 ◽  
Author(s):  
David Alon Rose
Keyword(s):  
Compact Space ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Compact ◽  
Weakly Continuous ◽  
Countable Space ◽  
Second Countable Space ◽  
Graph Function ◽  
Almost Continuous

AbstractA strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

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.

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

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.

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

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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