First category function spaces under the topology of pointwise convergence

Abstract

A large class of function spaces, under the topology of pointwise convergence, are shown to be of first category. The question as to when a space of continuous functions is of first category (i.e., can be written as a countable union of nowhere dense subsets) seems to be relatively unanswered. When the topology on the function space is the supremum metric topology, then if the range space is completely metrizable, so is the function space. Thus by the Baire category theorem, there is a large class of function spaces having the supremum metric topology which are Baire spaces (i.e., no open subspace is of first category). However, an example is given in L 4] of a metrizable Baire space Y such that the space of continuous functions from I, the closed unit interval, into Y, under the supremum metric topology, is of first category. If the domain space is compact, the supremum metric topology agrees with the compact-open topology on the function space, so there is also a large class of function spaces having the compact-open topology which are Baire spaces. However, the situation changes dramatically when the topology of pointwise convergence is imposed on the function spaces. Under this topology, the function spaces are of first category for most nonpathological domain and range spaces. For example, it will follow from the Theorem in this paper that the space of real-valued continuous functions on /, with the topology of pointwise convergence, is of first category. The notation C (X, Y) will stand for the space of all continuous functions from X into Y under the topology of pointwise convergence. This topology is generated by the base $ = I H L*;. V.]|x. e X and V. is open in YK Presented to the Society, January 23, 1975; received by the editors April 13, 1974. AMS (MOS) subject classifications (1970). Primary 54C35; Secondary 54D99.