Extension dimension and quasi-finite CW-complexes

Abstract

We extend the definition of quasi-finite complexes by considering not necessarily countable complexes. We provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Further, we extend the definition of U V (L)-spaces on non-compact case and show that some properties of U V (n)-spaces and U V (n)-maps remain valid, respectively, for U V (L)-spaces and U V (L)-maps.

Universal absolute extensors in extension theory

“…It follows from [8,Corollary 2.2] that none of the Eilenberg-MacLane complexes K(G, n), n ≥ 2 and G an Abelian group, is quasi-finite. Therefore Theorem 3.5 implies the following result.…”

“…A complex L possesses the connected pairs property with respect to Polish spaces iff L is quasi-finite.Proof. The "if" part follows from[8, Proposition 2.4]. In order to establish the "only if" part we shall show that L satisfies property (d) from Theorem 2.3.…”

Locally n-Connected Compacta and UVn -Maps

“…The next theorem follows from a stronger result due to Pasynkov [15,Theorem 6], and its proof is based on Dranishnikov results [6, Theorem 1] and [7, Theorem 1.2] (such a result concerning the extension dimension with respect to quasi-finite complexes was established in [14,Proposition 2.7]). We provide here a proof based on factorization theorems.…”

Compact maps and quasi-finite complexes

“…We introduce various notions of quasi-finite families and discuss relations among them. We also generalize the notion of a K-invertible map (see [17]) for a CW complex K to an F -invertible map where F is a family of maps between CW complexes. We discuss relations between existence of invertible maps and quasi-finite families.…”