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I have a question about terminology. Suppose we have a space X with the property that:

for all x, x' in X and neighborhood N of x, N is homeomorphic to some neighborhood N' of x'

OR

for all x, x' there exists a homeomorphism f:X→X s.t. f(x)=x'.

(I believe these are equivalent, but I haven't worked it out.) In some sense, these spaces are uniform (although I know that uniform space has its own meaning). There are no "distinguished" points, or different "types" of points. (Any open, simply-connected subset of Euclidean space has this property. Any closed subset of Euclidean space not equal to its boundary lacks it, since boundary points cannot be continuously mapped onto interior points.)

Is there a name for this?

EDIT: fixed an error.

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# Uniformity of space

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