A near-selection theorem is proven for carriers defined on spaces that are the countable union of finite dimensional compacta. As an application a new proof is given of the fact that the space of homeomorphisms on a compact piecewise linear manifold is locally contractible. In addition a new criterion is given to determine if the space of homeomorphisms on a compact n-manifold is an l2-manifold.
In this paper we introduce the concept of Δ-maximal topologies and characterize Δ-maximal topological spaces for a class of topological properties which include the cardinal functions κ-dense, Sanin, Caliber-κ, and the countable chain condition. In addition special consideration is given to the existence of Δ-maximal κ-dense spaces which are finer than a given κ-dense topology and also investigated are the existence of Δ-maximal second countable spaces.
Nearnesses of finite order are defined, and in particular a sub-class (called Ivanov n-nearnesses) which includes the class of Lodato proximities and leads to contiguities as a limiting case. A canonical compactification by maximal clans is constructed and characterised descriptively, thus unifying the compactification theories of I.O-proximities and of contiguities. Finally, it is shown how all maximal clans with respect to the finest contiguity compatible with a given Ivanov n-nearness can be constructed from prime closed filters, using not more than n such filters for each clan.
This paper studies the relationships of θ-spaces to other generalized metric spaces. In particular, a condition shared by θ-spaces and spaces with a quasi-Gg diagonal is introduced, and it is shown that every regular θ refinable β-space satisfying this condition is semi-stratifiable. In addition, a regular quasi-complete space satisfying this condition has a base of countable order.
A complete survey of results on the epimorphism problem for various categories of topological groups is given. All known partial results on the epimorphism problem for Hausdorff topological groups support the conjecture that epimorphisms of Hausdorff topological groups have dense image.
Sufficient conditions are given for the union of two Hilbert cube (manifolds) intersecting in a Hilbert cube (manifold) to be a Hilbert cube (manifold). The corollaries include a non-stabilized mapping cylinder theorem for embeddings between Hilbert cube manifolds and a sum theorem for Keller cubes.
In this paper it is shown that if X is a connected space which is not pesudocompact, then βX is not movable and does not have metric shape. In particular βX cannot have trivial shape. It is also shown that if X is Lindelöf and KχβX−X is a continuum, then K cannot be movable or have metric shape unless it is a point.
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