Suppose a topological space X is locally Euclidean. Indeed, there is a homeomorphism $\varphi_{1}$ from the open set $U_{1}$ in X to an open set $V_{1}$ in $\mathbb{R}^{n}$
This is called an open chart of any points x contained in $U_{1}\subseteq \mathbb{R}^{n} \in X$.
If another open chart of x exists then the geometric interpretation looks like "image below".
I tried 'sketching' it as to how it looks like. I hope it is correct as my doubts are predicated on the correctness of the image.
Recall: A topological space is Locally Euclidean if $\forall x \in X, \exists$ open set $U \subseteq X$: U is homeomorphic to some open set $V \subseteq \mathbb{R}^{n}, \exists n \in \mathbb{Z}^{+}$
Clarifications at this point would be helpful.
Clearly, $U_{1} \cap U_{2} \subseteq U_{1}$ and $U_{1} \cap U_{2} \subseteq U_{2}$. $U_{1}$ is homeorphic to $V_{1}$ and $U_{2}$ is homeomorphic to $V_{2}$.
Are the two $\mathbb{R}^{n}$ space different? My thought on this is they are since there can exists many open sets U in X and by the definition of locally euclidean, it suffices only that every element in X belongs to some open set. From this, each open set is homeomorphic to some open set V in $\mathbb{R}^{n} \exists$ n $\in \mathbb{Z}^{+}$