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CompactT 0-spaces andT 0-compactifications

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This paper deals with compactifications, particularly withk-completions, ofT 0-spaces.

The concept ofk-complete spaces, wherek is a cardinal with 1⩽k ⩽ ω, provides a useful gradation of compactness. The compact spaces are precisely the ω-complete spaces. For smallerk one obtains more restrictive concepts. However, for Hausdorff spaces, these concepts coincide for allk⩾2. For eachk, a space$ k is constructed such that thek-completeT 0-spaces are precisely the extension closed subspaces of powers of$ k . Fork⩾2 the associated$ k -compactificationsX\(\beta _{\$ _k } \) X areC*-embeddings. More generally, for a finite spaceE, allE-compactificationsX ↪ β E X areC*-embeddings if and only if the Sikorski spaces$ 2 is a quotient of some connected closed subspace ofE.

The main contribution of this paper is a construction, which associates, for anyk, with anyT 0-spaceX a minimalk-completion β k X. For normal spacesX, the extensions β2 X, ..., βω X coincide with the Čech-Stone compactification βX ofX. ForT 1-spacesX, the extension βω X coincides with the Wallman compactification ωX ofX. If β k X\X is finite (which happens, e.g., ifk=1 or ifX is finite), then β k X is an almostk-complete reflection ofX. However, for eachk⩾2, thek-complete spaces are not almost reflective inTop. They are, however, implicational inTop and thus expressible as an intersection of almost reflective subcategories ofTop.

In addition the concept of nearly closed subspaces is introduced, coinciding for Hausdorff spaces with that of closed subspaces, such that each of the following conditions characterizes compactness ofT 0-spacesX:

  1. (a)

    X is a nearly closed subspace of a product of finiteT 0-spaces

  2. (b)

    X is a nearly closed subspace of a power of the Sierpinski space$ 1

  3. (c)

    X is nearly closed in every extension ofX

  4. (d)

    for each spaceY, the projectionX ×YY preserves near closedness.

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Herrlich, H. CompactT 0-spaces andT 0-compactifications. Appl Categor Struct 1, 111–132 (1993).

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