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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The closed image of a metrizable space is $ M\sb{1}$


Author: F. G. Slaughter
Journal: Proc. Amer. Math. Soc. 37 (1973), 309-314
MSC: Primary 54D15
MathSciNet review: 0310832
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Abstract: J. Ceder introduced the notions of $ {M_1}$ space (a regular space with $ \sigma $-closure preserving base) and stratifiable space as natural generalizations of Nagata and Smirnov's conditions for the metrizability of a regular space. Even though a topological space Y which is the image of a metrizable space under a closed, continuous mapping need not be metrizable, we show as our main result that Y will have a $ \sigma $-closure preserving base. It follows that one cannot obtain an example of a stratifiable space which is not $ {M_1}$ by constructing a quotient space from an upper semicontinuous decomposition of a metric space. In the course of establishing our major result, we obtain conditions under which the image of certain collections of sets under a closed, continuous mapping will be closure preserving.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310832-X
Keywords: space, $ \sigma $-closure preserving base, closed mapping, Lašnev space
Article copyright: © Copyright 1973 American Mathematical Society