Abstract
A first-countable space is called maximal if it is not contained as a dense subspace in a first-countable space properly. The following are shown; (1) every locally compact, first-countable space is a dense subspace of a maximal space, (2) every metrizable space is a dense subspace of a maximal space, and (3) there is a first-countable space which is not a dense subspace of any maximal space.
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