Uniformity of Space: Definition & Examples

[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.

 

[NeroKid]

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

 

[alexfloo]

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

 

[micromass]

Continuous bijection should be sufficient:

Continuous X→Y means for each open O_Y subset of Y, f^-1(O_Y)=O_X is open in X. Bijection means that

f(O_X)=f(f^-1(O_Y))=O_Y,

so f^-1 is also continuous.

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

 

[blue_raver22]

This property is known as uniformity of space or uniform continuity. It means that the space does not have any distinguishing features or singularities, and all points are equivalent in terms of their local neighborhoods. This is a desirable property in many mathematical and scientific contexts, as it allows for simpler and more uniform analysis and understanding of the space. However, it is important to note that this is not the same as a uniform space, which has a more specific definition in mathematics.