Cauchy sequences in semimetric spaces

Burke, Dennis K.

doi:10.1090/S0002-9939-1972-0290328-3

Cauchy sequences in semimetric spaces
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Proc. Amer. Math. Soc. 33 (1972), 161-164 Request permission

Abstract:

As the main result we prove that every semimetrizable space has a semimetric for which every convergent sequence has a Cauchy subsequence. This result is used to show that a ${T_1}$ space X is semimetrizable if and only if it is a pseudo-open $\pi$-image of a metric space.

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