Compactness and the Axiom of Choice

Abstract

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. 1. C-compact spaces from the eprireflective hull in Haus of C-compact completely spaces. 2. Equivalent are: (a) the axiom of choice, (b) A-compact = D-compactness. (c) B-compactness = D-compactness, (d) C-compactness = D-compactness and complete regularity, (e) products of spaces with finite topologies are A-compact,> (f) products of A-compact spaces are A-compact , (g) products of D-compact spaces are D-compact, (h) powers X k of 2-point discrete spaces are D-compact, (i) finite products of D-compact spaces are D-compact, (j) finite coproducts of D-compact spaces are D-compact, (k) D-compact Hausdorff spaces form an epireflective subcategory of Haus, (l) spaces with finite topologies are D-compact. 3. Equivalent are: (a) the Boolean prime ideal theorem, (b) A-compactness = B-compactness, (c) A-compactness and complete regularity = C-compactness, (d) products of spaces with finite undelying sets are A-compact, (e) products of A-compact Hausdorff spaces are A-compact, (f) powers X k of 2-point discrete spaces are A-compact, (g) A-compact Hausdorff spaces form an epireflective subcategory of Haus. 4. Equivalent are: (a) either the axiom of choice holds or every ultrafilter is fixed, (b) products of B-compact spaces are B-compact. 5. Equivalent are: (a) Dedekind-finite sets are finite, (b) every set carries some D-compact Hausdorff topology, (c) every T 1 has a T 1 — D-compactification, (d) Alexandroff-compactifications of discrete spaces are D-compact.