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

TL;DR

This paper proves that a path-connected, locally direct product, closed set in a topological product space is globally a direct product, with implications for fiber spaces and topological manifolds.

Contribution

It establishes that local direct product conditions imply a global product structure for certain topological sets, providing an elementary proof and discussing broader implications.

Findings

Local direct product sets are globally direct products.

Example of a manifold not embeddable as a submanifold in a direct product.

Introduction of topological 2-space and related homotopy concepts.

Abstract

Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\in X\times Y$, 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 $(a,b) \in U\times V$ and $$C\cap (U\times V)=I\times J$$ 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 $$C=A\times B$$ 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 $C([0,1];\mathbb R)$-manifold that cannot be embedded in the direct product $(C([0,1];\mathbb R))^n$ as a $C([0,1];\mathbb R)$-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. Finally, we raise a question on the existence of an $\mathbb R^n$-Morse function and the existence of an $\mathbb R^n$-immersion in a finite-dimensional $\mathbb R^n$-Euclidean space. Here, we note that the problem of defining the concept of an $\mathbb R^n$-handle body may also be considered.