In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable and locally finite CW complex L the following conditions are equivalent:-L is quasi-finite.-There exists a [L]-invertible mapping of a metrizable compactum X with e − dim X ≤ [L] onto the Hilbert cube. Finally, we construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex.
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.
We say that a C*-algebra X has the approximate n-th root property (n ~£ 2) if for every a € X with ||a|| < 1 and every e > 0 there exits b 6 X such that ||6|| < 1 and \\a -b n \\ < t. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (respectively, commutative) separable unital C-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Cech cohomology.
Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.
Abstract. Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension ≤ [L] contains a universal element which is an absolute extensor in dimension [L]. Our main result shows that L is quasi-finite.
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