General Topology and its Applications最新文献

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.

The first part of this paper supplements earlier work of the first-named author by exhibiting an example of a wild n-sphere Σ in Eⁿ⁺¹ (n⩾4) for which each horizontal n-dimensional hyperplane of Eⁿ⁺¹ that does meet Σ intersects it either in a point or in an (n − 1)-sphere that is flatly embedded in the hyperplane. The second part sets forth an improved criterion for detecting the n-cell (n ≠ 4) based upon properties of slices determined as inverse sets associated with maps of a space to an interval.

This paper presents an example of a shrinkable (in the sense of Bing) cellular upper semi-continuous decomposition of a non-metric Hausdorff 2-manifold V such that the associated decomposition space is a 2-manifold topologically distinct from V.

An initial investigation in $\text{F}$-expansion relative to families of continuous functions on the acting group is presented, modelled after previous work of Bowen and Walters on real flows. Basic properties are established, and expansion in the natural class of non-trivial homomorphisms is extensively studied. Finally, modelling discrete flows with such expansion in symbolic subshifts is investigated. Generalizations to Rⁿ and Zⁿ are indicated.

In this paper we investigate the topological structure of the Graev free topological group over the rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generated by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some properties of certain canonical maps which are closely related to the topological structure of free groups.