On intersections of compacta in Euclidean space: the metastable case

Abstract

We prove the following theorem: Let $f:X\rightarrow R^{n}$ and $g:Y\rightarrow R^{n}$ be any maps of compacta $X$ and $Y$ into the Euclidean $n$-space $R^{n}$, $n\geqq 5$. Suppose that $\dim(X\times Y) \lt n$ and that 2 $\dim X+$$\dim Y \lt 2n-1$. Then for every $\epsilon \gt 0$ there exist maps $f^{\prime}$:$X\rightarrow R^{n}$ and $g^{\prime}$:$Y\rightarrow R^{n}$ such that $d(f, f^{\prime}) \lt \epsilon,$d(g, g^{\prime}) \lt \epsilon$ and $f^{\prime}(X)()g^{\prime}(Y)$$=\emptyset$.