A note on dimension theory of metric spaces
A note on D-spaces
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.
A note on -metrizable spaces
In this paper, the relationships between metric spaces and -metrizable spaces are established in terms of certain quotient mappings, which is an answer to Alexandroff’s problems.
A note on k-c-semistratifiable spaces and strong -spaces
Recall that a space is a c-semistratifiable (CSS) space, if the compact sets of are -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a -space is a k-c-semistratifiable space if and only if has a function which satisfies the following conditions: (1) For each , and for each . (2) If a...
A note on maximally resolvable spaces.
A note on metrics and tolerances
A note on monotone countable paracompactness
We show that a space is MCP (monotone countable paracompact) if and only if it has property , introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally countably compact closed mappings, from which it follows that both strongly Fréchet spaces and q-space closed images of MCP spaces are MCP. Some results on...
A note on multivalued Baire category theorem
A note on operators extending partial ultrametrics
We consider the question of simultaneous extension of partial ultrametrics, i.e. continuous ultrametrics defined on nonempty closed subsets of a compact zero-dimensional metrizable space. The main result states that there exists a continuous extension operator that preserves the maximum operation. This extension can also be chosen so that it preserves the Assouad dimension.
A note on pairwise c-compact bitopological spaces
A note on products of Fréchet spaces
A note on Raghavan-Reilly's pairwise paracompactness.
A note on sequence-covering -images of metric spaces
A note on some fixed point results.
A note on -spaces and -metrizable spaces
In this paper, we give the mapping theorems on -spaces and -metrizable spaces by means of some sequence-covering mappings, mssc-mappings and -mappings.
A note on star Lindelöf, first countable and normal spaces
A topological space is said to be star Lindelöf if for any open cover of there is a Lindelöf subspace such that . The “extent” of is the supremum of the cardinalities of closed discrete subsets of . We prove that under every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under , which shows that a star Lindelöf, first countable and normal space may not have countable extent.
A note on the comparison of topologies.
A note on the Hurewicz isomorphism theorem in Borsuk's theory of shape
A note on the invariance of Baire spaces under mappings
A note on the non-emptiness of the limit of approximate systems
Short proofs of the fact that the limit space of a non-gauged approximate system of non-empty compact uniform spaces is non-empty and of two related results are given.