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I want to know the example which satisfies the following.

X is topological space.

for every point x,y in X, there exist open nbhd Ux,Uy of x,y which are homeomorphic.

X has some kind of good conditions, i.e hausdorff,locally connected,locally compact, 2nd countable.. etc..

X is not locally Euclidean .

I think.... the thing like topologically homogeneous space which is not manifold. but I can't find the good example.

In fact, another goal of my question is this. "How can I make locally euclidean property from the other topological properties."

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topologically homogeneous space?

I want to know the example which satisfies the following.

X is topological space.

for every point x,y in X, there exist open nbhd Ux,Uy of x,y which are homeomorphic.

X has some kind of good conditions, i.e hausdorff,locally connected,locally compact, 2nd countable.. etc..

X is not locally Euclidean .

I think.... the thing like topologically homogeneous space which is not manifold. but I can't find the good example.