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, F∗X and αX, of
such a space are described, and it is shown that if ShpX =ShpY , then
ShpF∗X =ShpF∗Y ; 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.
