General Topology and its Applications最新文献

We correct an error in the above-mentioned paper and provide a solution to an open problem contained therein.

A space X partitions a space Y if Y is the union of pairwise disjoint subjets, each of which is homeomorphic to X. We study the topological partition relation, particularly in the context of separable metric spaces, obtaining topological analogues to well-known problems in the theory of geometric partitions.

Let f:S³→S² be a continuous function. If yϵS² assume that f^-1 (y) has the shape of a circle and that there are neighborhoods V ⊂ U of f^-1(y) such that for any point inverse f^-1(z)⊂V, the inclusion of f^-1(z) into U is essential. We show that f can be approximated arbitrarily closely by Seifert fiber maps.

Good ultrafilters produce topological ultraproducts which enjoy a strong Baire category property (depending upon how good the ultrafilter is). We exploit this property to prove a “uniform boundedness” theorem as well as a theorem which says that, under the Generalized Continuum Hypothesis (GCH), many ultraproduct spaces have families consisting of closed discrete sets of high cardinality such that every nonempty open set contains one of these sets. In another section we relate the strong Baire properties to the infinite distributivity of Boolean Algebras of regular open sets. Finally, we prove that, under the GCH, a great many topological ultrapowers are homeomorphic to the corresponding ultrapower of the space of rational numbers; and we show further that the GCH is indispensable to the proof. A purely model-theoretic application of our methods solves a problem related to the Keisler-Shelah Ultrapower Theorem.

In this paper the lattice of all epireflective subcategories of a topological category is studied by defining the T₀-objects of a topological category. A topological category is called universal iff it is the bireflective hull of its T₀-objects. Topological spaces, uniform spaces, and nearness spaces form universal categories. The lattice of all epireflective subcategories of a universal topological category splits into two isomorphic sublattices. Some relations and consequences of this fact with respect to cartesian closedness and simplicity of epireflective subcategories are obtained.

In this paper we give characterizations of some classes of compact topological spaces, such as (products of) compact lattice, tree-like and orderable spaces, by means of the existence of a closed subbase of a special kind.

This paper gives conditions under which the inverse limit of a system of compact (but non-Hausdorff) spaces will be non-empty, or compact, or hereditarily compact. The main result (Theorems 3 and 5) is that, if the spaces are compact, T₀ and non-empty and the maps are closed and continuous, then the inverse limit is compact and non-empty (and, trivially, T₀). Simple examples are given to show that the results are reasonably sharp.

In the first paragraph we study filters in the lattice I^X, where I is the unitinterval and X an arbitrary set. The main result of this section is a characterization of minimal prime filters in I^X containing a given filter in I^X by means of ultrafilters on X.

In the second paragraph we apply the results of the previous section to define convergence in a fuzzy topological space which enables us to characterize fuzzy compactness and fuzzy continuity.

Many of earlier and recent results obtained for p-spaces and their relatives can be extended by using a simple and natural concept (relative compactness) which was defined and investigated in an earlier paper of the author.

In the present paper extensions of recent metrization theorems concerning p-spaces are dealt with. A recent metrization theorem of J. Nagata is extended to relative compactness. Under the assumption of the continuum hypothesis A.V. Arhangel'skiǐ's problem asking whether a space, each subspace of which is a paracompact p-space, contains a dense metrizable subspace, is solved affirmatively (and for the generality of relative compactness). Some results concerning the behaviour of the first axiom of countability and its generalizations under relative compactness are also included.

Let X be an arbitrary product of separable complete metric spaces. It is proved that every automorphism of the “category algebra” (the Baire sets modulo first category sets) of X can be obtained from some one-to-one map T of X onto itself such that both T and T⁻¹ preserve Baire sets and first category subsets of X.