In this paper, for a 3-dimensional complete submanlfold $M$ with parallel mean curvature vector in $S^{3+p}(c)$ , we give a pinching condition of the Ricci curvature under which $M$ is a 3-dimensional small sphere.

This paper contains two main results. One is to prove the orthocompactness of products with a metric-like factor. Another is to characterize the orthocompnct products of spaces of ordinals. In particular, from this characterization, we can obtain the equivalence with the normality of these products.

We show that any topological pair with normally embedded subspace has the strong shape of a pair, such that the inclusion map of the subspace into the total space is a cofibration. Furthermore we prove that a strong shape morphism of pairs is a strong shape equivalence if and only if it operates as strong shape equivalence of the total spaces and of the subspaces considered separately.

Let $C_{k}(X)$ be the function space with the compact-open topology over a Tychonoff space $X$ and $\xi$ a continuous real-valued function on $C_{k}(X)$. A closed subset $S$ of $X$ is called a support for $\xi$ if $\xi(f)=\xi(g)$ holds for any pair $(f, g)$ of functions in $C_{k}(X)$ such that $f|_{s}=g|_{s}$. It is proved that the minimal support for any realvalued continuous function on the space $C_{k}(X)$ exists.