C-14 - Spaces of Functions in Pointwise Topology

Abstract

If X is a space, RX is the topological product of X copies of the usual space R of real numbers. Elements of RX can be interpreted as real-valued functions on X. The topology of the space RX is called the topology of pointwise convergence, or the pointwise topology. The set C(X) of all real-valued continuous functions on X is a dense subset of RX. The topology of RX generates the subspace topology on C(X). Cp(X), the space of real-valued continuous functions on X in the pointwise topology, is obtained by endowing C(X) with the topology of a subspace of the space RX. Spaces Cp(X) are also called Cp-spaces. Investigation of topological properties of function spaces originated in functional analysis. One of the central positions in functional analysis is occupied by the concept of the weak topology of a Banach space. These topologies produce a natural strata of non-metrizable topological spaces. One of the implications is that the topological properties of Banach spaces with weak topologies can vary along a wider scale than the topological properties of Banach spaces with standard (metrizable) topology.