Let $Y$ be a topological space, and $M$ be second countable Hausdorff topological space such that for any $p\in M$ it holds that there is an open neighborhood $U(p)$ in $M$ such that $U(p)\approx Y$ where with $\approx$ the homeomorphism relation is denoted. I wonder what $M$ can look like if $Y$ is different from $\mathbb R^n$.
First of all I was thinking about $Y = \mathbb S^1$ but then realized that taking $Y$ compact is quite restricting. Indeed, if $Y$ is Hausdorff second countable and compact then for any $p\in M$ there is a compact neighborhood $U(p)$ and since $M$ is Hausdorff, $U(p)$ is closed and open and hence is a union of connected components of $M$ which is homeomorphic to $Y$. If I am not wrong, it means that $M$ is a disjoint union of $Y$, i.e. $ M\approx \coprod\limits_{i\in I}Y_i $ for some countable (i.e. finite of countably infinite) index set $I$. Please correct proof if it's wrong.
On the other hand, if $Y$ is Hausdorff and second countable but not compact, the space $M$ may be more interesting, i.e. for $Y$ being $C\cap (0,1)$ with $C$ - Hausdorff set or $Y$ being torus without one point. I wonder if there are any spaces of interest which are Hausdorff second countable but not locally Euclidean, being rather locally homeomorphic to some more complicated spaces.