Functional product topology: A common framework of the topological sum, the product topology and the functional topology of pointwise convergence

Based on the topological sum, for X being a discrete topological space, we prove that the product topology generated by some topological spaces is equal to the topology of pointwise convergence related to X. For every topology on a Cartesian product, we find an equivalent condition under which every projection from the product to each component is continuous. We propose the definition of functional product spaces, which can be considered as a common framework of the product spaces, the topological sums and the functional spaces (the topology of pointwise convergence).