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."

wouldbe called a manifold in the literature. Take a google scholar search of "fractal manifold". Things that look locally like a Sierpinski gasket would be Sierpinski gasket manifolds. FYI it doesn't take much to make the Sierpinski space into a locally homogeneous space I believe removing the three extremal vertices should do the job. $\endgroup$ – Ryan Budney Aug 24 '10 at 17:53