Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that:

$F$ separates points of $X$, i.e. for any two distinct points $x,y\in X$ there is $\varphi\in F$ such that $\varphi(x)\ne\varphi(y)$;

for every $x\in X$ there is an open neighbourhood $U$ of $x$ and $\varphi\in F$ such that $\varphi|_U$ is a homeomorphism of $U$ onto an open set $W$ in $\mathbf{R}^d$.

Does it follow that $F$ generates the topology of $X$, i.e. the original topology of $X$ is the minimal topology in which all elements of $F$ are continuous?

Since $F$ consists of continuous maps, it generates the topology weaker than the original one, and since $F$ separates points, the generated topology is Hausdorff.

It seems that this topology is also locally Euclidean, but I cannot articulate the right reasoning. Then we get that the identity is a continuous bijection between the original and generated topologies, and since they both are locally Euclidean it follows that this map is open, and therefore a homeomorphism. However, I feel concerned about the very last argument, since it does not use Hausdorff property, without which the statement is wrong.

PS You may assume that all the objects considered are actually smooth or holomorphic, but I don't think it is relevant here.