  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.
