Every semi-stratifiable space or strong -space has a -cushioned (mod)-network. In this paper it is showed that every space with a -cushioned (mod)-network is a D-space, which is a common generalization of some results about D-spaces.
An existing description of the cartesian closed topological hull of , the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of , the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of...
For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems...
We show that a completely regular space Y is a p-space (a Čech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.
In this paper we improve some mapping theorems on -spaces. For instance we show that an -space is preserved by a closed and countably bi-quotient map. This is an improvement of Yun Ziqiu’s theorem: an -space is preserved by a closed and open map.
The main purpose of this paper is to establish general conditions under which -spaces are compact-covering images of metric spaces by using the concept of -covers. We generalize a series of results on compact-covering open images and sequence-covering quotient images of metric spaces, and correct some mapping characterizations of -metrizable spaces by compact-covering -maps and -maps.
This paper deals with the behavior of -spaces, countably bi-quasi--spaces and singly bi-quasi--spaces with point-countable -systems. For example, we show that every -space with a point-countable -system is locally compact paracompact, and every separable singly bi-quasi--space with a point-countable -system has a countable -system. Also, we consider equivalent relations among spaces entried in Table 1 in Michael’s paper [15] when the spaces have point-countable -systems.
We characterize the subsets of the Alexandroff duplicate which have a G-diagonal and the subsets which are M-spaces in the sense of Morita.