When I study topological manifold, I think some property of manifolds are so important that they can "almost characterize" manifolds. But I know a topological manifold is not easily to be characterized because of its locally Euclidean property.
I want to find a "weird" example of a locally compact, locally connected, Hausdorff and second countable space $X$ that is "nowhere locally Euclidean", i.e. for any point in $X$, there doesn't exist an open neighborhood that is homeomorphic to a open subset of a Euclidean space.
Since a discrete space is a 0-dimensional manifold(locally $\mathbb R^0$), the following type of example is not what I want:
A straight line union a point which is not on that line
Does such an example exist?
Every solution or reference or even more generalized example(Even stronger conditions can't guarantee the locally Euclidean property) will be appreciated!