Now I wonder if there is a space which is not locally Euclidean. eg. no matter how small piece you take, it's still not flat.
[Curved spaces are nowhere locally Euclidean. An example of a space that is locally Euclidean in some places but not others would be a line segment. The body of the line segment is locally Euclidean, but the endpoints are not: In Euclidean space, you can always turn 180 degrees and walk the other way, but at the endpoint, you can walk in one direction but not the other. -Raymond]