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Compactification and completion as absolute closure

Author: Anthony W. Hager
Journal: Proc. Amer. Math. Soc. 40 (1973), 635-638
MSC: Primary 54D35
MathSciNet review: 0331326
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Abstract: It is well known that a Tychonoff (respectively, Hausdorff uniform) space is compact (resp., complete) iff it is absolutely closed, i.e., dense in no other such space; we shall sketch a proof of this to our purpose, which is: Given $ X$, we apply Zorn's Lemma to obtain a space maximal with respect to the property of containing $ X$ densely, thus compact or complete. For uniform spaces, the ``maximal extension'' is automatically the completion; for Tychonoff spaces, we must, and do, explicitly arrange things so that the ``maximal extension'' has the universal mapping property describing the Stone-Čech compactification. A variant of the construction yields the Hewitt realcompactification. A crucial point in the proofs is (of course) the exhibition of an upper bound for a chain; this is, in essence, a direct limit construction.

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Keywords: Compactification, completion, absolutely closed
Article copyright: © Copyright 1973 American Mathematical Society