Vol. 84, No. 2, 1979

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Quasicompactifications and shape theory

Billy Joe Ball

Vol. 84 (1979), No. 2, 251–259

If f : X Y is an embedding of a space X into a space Y such that each component of Y is a compactification of the image of a quasicomponent of X and such that f induces a homeomorphism of the space QX of quasicomponents of X onto the space of components of Y , then (f,Y ) is called a quasicompactification of X. After some preliminary results, it is shown that a locally compact metric space X has a locally compact metric quasicompactiflcation if and only if QX is locally compact. Two canonical quasicompactifications, FX and αX, of such a space are described, and it is shown that if ShpX = ShpY , then ShpFX = ShpFY ; the question whether also ShpαX = ShpαY is left open. Finally, some techniques of this paper are used to obtain a proper shape version of a theorem due to Y. Kodama, generalizing previous work of the author.

Mathematical Subject Classification 2000
Primary: 54B15
Secondary: 54C56
Received: 30 January 1979
Published: 1 October 1979
Billy Joe Ball