Vol. 15, No. 1, 1965

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
Concerning Koch's theorem on the existence of arcs

L. E. Ward

Vol. 15 (1965), No. 1, 347–355
Abstract

A theorem of R. J. Koch asserts that if X is a compact space endowed with a partial order Γ such that

(i) Γ is a closed subset of X × X,

(ii) there exists 0 X such that (0,x) Γ for each x X, and

(iii) for each x X the set L(x) = {y : y x} is connected, then each point of X lies in a connected chain containing 0. In particular, X is arcwise connected. This is a corollary of the theorem: if X is a compact space and Γ is a partial order satisfying (i), and if W is an open subset of X such that each neighborhood of each point x of W contains a point yx with (y,x) Γ, then each point of W is the supremum of a connected chain which meets X W. A new proof of these results is presented.

The first of these theorems is generalized in several ways. The compactness is relaxed to local compactness and the assumption that each closed chain has a zero. Moreover, the existence of a zero need not be assumed. If the set E of minimal elements is closed, then E is joined by connected chains to all other points of X. If the set function L is continuous, then E is necessarily closed.

Mathematical Subject Classification
Primary: 54.55
Milestones
Received: 13 February 1964
Published: 1 March 1965
Authors
L. E. Ward