A universal complex structure on complete metrizable topological spaces

Let X be a topological space whose topology may be defined by a complete metric d. Taking all such metrics d we define a universal complex structure on X. For this complex structure the sheaf of germs of holomorphic functions on X coincides with the sheaf of germs of continuous functions on X, and hence the theories of topological and holomorphic vector bundles on X are the same.     

Finite posets and topological spaces in locally finite varieties

We prove that if a finite connected poset admits an order-preserving Taylor operation, then all of its homotopy groups are trivial. We use this to give new characterisations of locally finite varieties omitting type 1 in terms of the posets (or equivalently, finite topological spaces) in the variety. Similar variants of other omitting-type theorems are presented. We give several examples of posets that admit various types of Taylor operations; in particular, we exhibit a topological space which is not an H-space but is compatible with a set of non-trivial identities, answering a question of W. Taylor.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form March 19, 2003.     

Recursive topological spaces

The research was supported by the Russian Foundation for Fundamental Research (Grant 093-01-01506).     

Supercomplete topological spaces

Supercomplete topological spaces and other variants of supercompleteness are defined in this paper. The main idea is to give different characterizations of Čech complete spaces. In particular it is shown that for paracompacta (and so for metrizable spaces) and topological groups the two notions of supercompleteness and Čech completeness coincide.      

Almost topological spaces

Summary Convergence spaces are studied from a geometrical point of view. In many cases we may construct a topological space VirtX, which can be considered as the resolution of the topological singularities ofX.

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Riassunto Spazi di convergenza vengono studiati da un punto di vista geometrico. In molti casi possiamo costruire uno spazio topologico VirtX che può essere considerato come la risoluzione delle singolarità diX.

     

Two topological properties of topological linear spaces

A topological and a geometrical-topological property, previously known only for normed linear spaces, are established here for much more general classes of topological linear spaces. This research was conducted at the University of Washington in 1963 when the first author was visiting there. The work of both authors was supported in part by the National Science Foundation, U. S. A. (NSF-GP-378).     

CLP-compactness for topological spaces and groups

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:

(i)

a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and

(ii)

a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).