Vol. 29, No. 1, 1969

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Critical points on rim-compact spaces

David Hilding Carlson

Vol. 29 (1969), No. 1, 63–65

In this note we prove that all the points of a rim-compact space X at which X is not locally compact are critical points for any local dynamical system defined on X. When a local system is global this result is obtained by extending the global system π on X to a global system ρ on the Freudenthal compactification Y of X, then showing that Y X is a critical set for ρ, and, finally, observing that Y X contains all the points of X at which X is not locally compact. This weaker result will appear in the author’s doctoral dissertation and requires the use of general extension theorems proven there. For this paper, we isolate those parts of the thesis which are pertinent to our theorem.

Mathematical Subject Classification
Primary: 54.82
Received: 1 July 1968
Published: 1 April 1969
David Hilding Carlson