An elementary approach to ‘Algebra ∩ topology = compactness’

✍ Scribed by G. Richter

• Publisher: Springer
• Year: 1996
• Tongue: English
• Weight: 226 KB
• Volume: 4
• Category: Article
• ISSN: 0927-2852
• DOI: 10.1007/bf00122689

✦ Synopsis

Herrlich and Strecker characterized the category Comp2 of compact Hansdorff spaces as the only nontrivial full epireflective subcategory in the category Top2 of all Hausdorff spaces that is concretely isomorphic to a variety in the sense of universal algebra including infinitary operations.

The original proof of this result requires Noble's theorem, i.e. a space is compact Hausdorff iff every of its powers is normal, which is far from being elementary. Likewise, Petz' characterization of the class of compact Hausdorff spaces as the only nontrivial epireflective subcategory of Top2, which is closed under dense extensions (= epimorphisms in Top2) and strictly contained in Top2 is based on a result by Kat~tov stating that a space is compact Hausdorff iff its every closed subspace is H-closed. This note offers an elementary approach for both, instead.