Product Topology

For two given topological spaces, it is natural to define a topology on the Cartesian product of the two underlying spaces. Motivated by the Euclidean plane viewed as the product of the real line by itself, the product topology is defined as the topology generated by the Cartesian product of open sets of each space. This chapter is devoted to relating the product topology with the topology of each one of the factors. Of special interest is the continuity of maps that have as domain or codomain a product space. It is also of interest what topological invariants are transferred from the factors to the product space. The concept of the product topology is generalized for an arbitrary number of topological spaces.