Consider the sphere $S^n$ and let $0 < \epsilon < 1$. We write
$$U_\epsilon:= \{x \in S_n: x_{n+1} \le \epsilon\}, \quad V_\epsilon: = \{x \in S^n: x_{n+1}\ge -\epsilon\}$$ Is it true that $U_\epsilon \cap V_\epsilon$ is homeomorphic to $S^{n-1}\times [-\epsilon , \epsilon]?$
I tried writing some obvious maps but nothing I came up works. Clearly it suffices to write down a continuous bijection because we work with compact spaces. If they are not homeomorphic, are they homotopic spaces?
The answer is yes. For a homeomorphism between the two spaces, consider the map $f: S^{n-1} \times [-\epsilon,\epsilon]\to U_{\epsilon} \cap V_{\epsilon}$ given by $$ f((x_1,\dots,x_n),t) = \left(\sqrt{1-t^2}\cdot x_1,\dots,\sqrt{1-t^2} \cdot x_n,t \right). $$