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.All spaces in this paper are Tychonoff unless otherwise specified. A space which contains a space A' as a dense subspace is called an extension of X. Let us call a first-countable space maximal, or, more precisely, maximal with respect to first-countability, if it has no proper, first-countable extension. According to [S, Theorem 2.9], a first-countable space is maximal if and only if it is pseudocompact. (Note that our maximal spaces are identical to Stephenson's first countableand completely regular-closed spaces.) Hence, if a first-countable space X has first-countable compactification Y, then Y is a maximal extension of X. On the other hand, even if X does not have a first-countable compactification, it can still have a maximal extension.Here, we are concerned with two questions. Namely, which first-countable spaces have maximal extensions, and whether all first-countable spaces have maximal extensions.In § §1 and 2, we shall answer the first question by showing that every locally compact, first-countable space and every metrizable space have maximal extensions.§3 will provide a negative answer to the second question.1. Locally compact, first-countable spaces. We shall begin withProposition. For a first-countable space X, the following conditions are equivalent: (l)X has a maximal extension,

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