Covariant and contravariant points of view in topology with applications to function spaces

Abstract

The purpose of this paper is to present a way of viewing of basic topology which unifies quite a few results and concepts previously seemed not related (quotient maps, product topology, subspace topology, separation axioms, topologies on function spaces, dimension, metrizability). The basic idea is that in order to investigate an unknown space X, one either maps known spaces to X or maps X to known spaces. Mapping known spaces to X leads to covariant functors. Therefore, it will be part of what we call the covariant point of view. Mapping X to known spaces leads to contravariant functors. It will be part of what we call the contravariant point of view. The covariant approach is an abstraction of the well known methodology of the homotopy theory: to investigate properties of CW complexes one computes their homotopy groups, i.e., one considers maps from spheres to CW complexes. Once some CW complexes are well understood, one can map them to a space X in order to detect its topological properties. The dual to covariant approach, the contravariant approach, is an abstraction of the well known methodology of the shape theory: to investigate topological properties of space X one maps X to CW complexes. It is explained in the paper that many notions/results can be better understood as analyzed from either covariant or contravariant points of view. Particular attention is given to function spaces. It is shown that the three main topologies on function spaces (the basic covariant topology, the compact-open topology, the pointwise convergence topology) can be introduced in the same manner: they are covariantly induced by functions f:S→Map(X,Y) so that adj X(f)|S×K is continuous for (a) K=X (the basic covariant topology), (b) any locally compact K in X (the compact-open topology), (c) any finite subset K of X (the pointwise convergence topology). By applying the concept of adjointness of functors, two new topologies on the product X×Y are introduced. The PC-product X× PC Y arises as a left adjoint to the pointwise convergence topology, and the CO-product X× CO Y arises as a left adjoint to the compact-open topology.