Locally Euclidean property in Topology

Question

I’m trying to understand the definition of locally Euclidean:

Definition: A topological space $X$ is locally Euclidean if there exists $n \in \mathbb{N}$ so that for all $x \in X$ there is a neighborhood $U$ of $x$ which is homeomorphic to an open subset $V \subset \mathbb{R} ^n$.

If $X$ and $Y$ are locally Euclidean, how would I show that $X \times Y$ is locally Euclidean? Also, how can we prove that the $n$-dimensional sphere $S^n$ is locally Euclidean?

I'm studying Introduction to Topology by Munkres.

Answer 1

If $X$ and $Y$ are locally Euclidean, how would I show that $X\times Y$ is locally Euclidean?

Let $(x,y)\in X\times Y$. By the assumption there is an open neighbourhood $U_x$ of $x\in X$ and a homeomorphism $f:U_x\to\mathbb{R}^n$ and there is an open neighbourhood $V_y$ of $y\in Y$ and a homeomorphism $g:V_y\to\mathbb{R}^m$. By the definition $U_x\times U_y$ is an open neighbourhood of $(x,y)\in X\times Y$ and we have a function

$$F:U_x\times V_y\to\mathbb{R}^n\times\mathbb{R}^m\simeq\mathbb{R}^{n+m}$$ $$F(a,b)=\big(f(a), g(b)\big)$$

All you need to show is that $F$ is a homeomorphism which I leave as an exercise.

Also, how can we prove that the $n$-dimensional sphere $S^n$ is locally Euclidean?

The stereographic projection gives an explicit homeomorphism between $S^n\backslash\{Q\}$ (which is open) and $\mathbb{R}^n$. You just need to take two of these with different base points in order to cover entire $S^n$.

Answer 2

For you first question, you should be aware that this is a LOCALLY property so that by choosing a smaller neighbourhood, you can assume V as a cube $[0,1]^n$. So that the product of two cube is again a cube.

For the second part, you can project the sphere to a plane in the space, this gives locally gives a homeomorphism.