Let $X$ be a set, $n\in\mathbb{N}$ and $((U_i,\phi_i))_i$ a family of subsets $U_i\subseteq X$ with injective functions $\phi_i: U_i\to\mathbb{R}^n$, which hold the following conditions:
$\bigcup_{i} U_i=X$
for every $i$ is $\phi_i(U_i)\subseteq\mathbb{R}^n$ open
for all $i,j$ is $\phi_i(U_i\cap U_j)\subseteq\mathbb{R}^n$ open and $\phi_j\circ\phi^{-1}_i:\phi_i(U_i\cap U_j)\to\mathbb{R}^n$ is smooth.
Show, that $X$ has exactly one topology, in regards to the family $((U_i,\phi_i))_i$ is an atlas.
Conclude: Is the family $(U_i.\phi_i)_i$ countable, then is $X$ with this atlas a smooth manifold.
Hello,
I Have a question to this task. How can I characterize the open subsets of $X$ with $(U_i,\phi_i)$ conveniently?
To get an atlas, it needs to be $\bigcup_{(V_i,\psi_i)} V_i=X$ with charts $(V_i,\psi_i)$.
The problem is, that $(U_i,\phi_i)$ are not charts, since $\phi_i$ is not a homeomorphism.
So I need a topology, such that $\phi_i$ is a homeomorphism, or not? How can I give such a topology?
Thanks in advance for your hints and comments.