A remark on locally direct product subsets in a topological Cartesian space

 Let X and Y be topological spaces. Let C be a path-connected closed set of X×Y. Suppose that C is locally direct product, that is, for any (a,b)∈X×Y, there exist an open set U of X, an open set V of Y, a subset I of U and a subset J of V such that (a,b)∈U×V and

C∩(U×V)=I×J

hold. Then, in this memo, we show that C is globally so, that is, there exist a subset A of X and a subset B of Y such that

C=A×B

holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. 

 In Appendix, we mentioned a simple example of a C([0,1];R)-manifold that cannot be embedded in the direct product (C([0,1];R))n as a C([0,1];R)-submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space.