Cofinitely and co-countably projective spaces

• Main Authors: Pablo Mendoza Iturralde, Vladimir V. Tkachuk
• Format: Article
• Language: English
• Published: Universitat Politècnica de València 2002-10-01
• Series: Applied General Topology
• Subjects: Cofinitely projective; Co-countably projective; Scattered compact
• Online Access: http://polipapers.upv.es/index.php/AGT/article/view/2062
• ISSN: 1576-9402; 1989-4147

We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.