Vol. 17, No. 1, 1966

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
On topologically induced generalized proximity relations. II

Michael Lodato

Vol. 17 (1966), No. 1, 131–135
Abstract

In the theory of proximity spaces of Efremovic, (The geometry of proximity, Mat. Sbornic, N.S. 31 (73), (1952), 189–200) the result: A set X with a binary relation “A close to B” is a proximity space if and only if there exists a compact Hausdorff space Y in which X can be imbedded so that A is close to B in X if and only if A meets B in Y (A denotes the closure of the set A) (Y. M. Smirnov, on proximity spaces, Mat. Sbornic, N.S. 31 (73), (1952), 543–574.) Raises the question: Can we display a set of axioms for a binary relation δ on the power set of a set X so that the system (X,δ) satisfies these axioms if and only if there is a topological space Y in which X can be imbedded so that

(1.1)  AδB in X if and only if A-∩ B-⁄= ϕ in Y.

In (M. W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. vol. 15, no. 3, June 1964, pp. 417–422), it is shown that an affirmative answer can be given if Y is T1 and if X is regularly dense in Y . The clusters of S. Leader, On clusters in proximity spaces, Fund. Math. 47 (1959), 205–213, were used in (M. W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. vol. 15, no. 3, June 1964, pp. 417–422). The present paper generalized this notion and thus relaxes the condition that X be regularly dense in Y . We actually characterize every system (X,δ) for which there exists a mapping f (not necessarily one-to-one) of X into a Hausdorff space Y such that

                          ---  ---
(1.2) A δB in X if and only ifAf ∩ fB ⁄= ϕ in Y.

Mathematical Subject Classification
Primary: 54.30
Milestones
Received: 26 September 1963
Published: 1 April 1966
Authors
Michael Lodato