I want to know an example of a topological space $X$ which satisfies the following.
for all points $x,y\in X$ and neighborhoods $U_x$, $U_y$, there exist neighborhoods $U'_x\subset U_x$, $U'_y\subset U_y$ of $x$ and $y$ that are homeomorphic (the homeomorphism does not have to map $x$ to $y$).
$X$ has some kind of good conditions, i.e Hausdorff, locally connected, locally compact, second countable, etc.
**X is not locally Euclidean ** (i.e., not a topological manifold)
I can't find the good example.
In fact, one goal of my question is this: "How can I make locally euclidean property from other topological properties."