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We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let F be the conjugate of a separable infinite-dimensional Banach space with bounded weak-* topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on F which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on F, F as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible. 0. Introduction. Let (X, x0) be a pointed topological space. For each integer n > 1, let Xn denote the cartesian product of n copies of X, and define in : Xn Xn+l by jn(xil . .., x) = (xi, ..., x, x0). Let (X, x0o)' denote the direct limit of the sequence

Contracting spaces of maps on the countable direct limit of a space