Properties of the Product of Compact Topological Spaces

One can prove the following proposition (1) For all topological spaces S, T holds Ω[: S, T :] = [:ΩS , ΩT :]. Let X be a set and let Y be an empty set. Note that [:X, Y :] is empty. Let X be an empty set and let Y be a set. Observe that [:X, Y :] is empty. We now state the proposition (2) Let X, Y be non empty topological spaces and x be a point of X. Then Y 7−→ x is a continuous map from Y into X↾{x}. Let T be a non empty topological structure. One can verify that idT is homeomorphism. Let S, T be non empty topological structures. Let us notice that the predicate S and T are homeomorphic is reflexive and symmetric. The following proposition is true (3) Let S, T , V be non empty topological spaces. Suppose S and T are homeomorphic and T and V are homeomorphic. Then S and V are homeomorphic.