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- Author or Editor: Shou Lin x
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A decomposition theorem about closed images of locally compact metric spaces is discussed. It is shown that a space is a closed image of a locally compact metric space if and only if it is a regular Fréchet space with a point-countable k-network, and each of its closed first-countable subset is locally compact.
Summary
We discuss the relationship between two different sequential connectedness, and prove that sequential connectedness is countably multiplicative.
In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space X is a metrizable space, if and only if X is a regular space with a σ-locally finite base at non-isolated points, if and only if X is a perfect space with a regular base at non-isolated points, if and only if X is a β-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in [7], which also shows that a space with a regular base at non-isolated points has a point-countable base.
The problem when a paratopolgical group (or semitopological group) is a topological group is interesting and important. In this paper, we continue to study this problem. It mainly shows that: (1) Let G be a paratopological group and put τ = ω H s(G); then G is a topological group if G is a P τ-space; (2) every co-locally countably compact paratopological group G with ω H s(G) ≦ ω is a topological group; (3) every co-locally compact paratopological group is a topological group; (4) each 2-pseudocompact paratopological group G with ω H s(G) ≦ ω is a topological group. These results improve some results in [11, 13].
In this paper, we characterize k-semi-stratifiable spaces by semi-continuous functions and give some applications. Also we give the similar characterizations of MCM spaces and K-MCM spaces.
