# Uniformity of space

1. Mar 22, 2012

### alexfloo

"Uniformity" of space

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.

Last edited: Mar 22, 2012
2. Mar 22, 2012

### NeroKid

Re: "Uniformity" of space

actually those 2 sentences are not equivalent since the continuous bijection doesnt guarantee the homeomorpic between neighborhood of x and x' , it must have continuous inverse to be homeomorphic

3. Mar 22, 2012

### alexfloo

Re: "Uniformity" of space

EDIT: You are correct. I'll add that in.

4. Mar 22, 2012

### micromass

Staff Emeritus
Re: "Uniformity" of space

Now you assume that all open sets are of the form $f^{-1}(O_Y)$. This is not necessarily true.