1

Disclaimer

This thread is meant to record. See: Answer own Question

Reference

It is a follow-up to: Uniform Spaces: Neighborhood System

It has relevance to: TVS: Uniform Structure

Problem

Given a topological space $\Omega$.

Consider inequivalent uniform structures: $\mathcal{U}\ncong\mathcal{U}'$

Can it happen that both induce the same topology: $\mathcal{U}^{(\prime)}\to\mathcal{T}$

Consider in particular TVS!

Community
  • 1
freishahiri
  • 17,045
  • I'm not quite sure what's going on here with this question, answer and (perhaps slightly condescending) comments. But if the purpose is to use the site to edify the readership, let me note that a much more interesting question has already been asked and answered here: http://math.stackexchange.com/questions/944416/is-there-always-an-equivalent-metric-which-is-not-complete. – Pete L. Clark Dec 21 '14 at 04:10
  • @PeteL.Clark: Mmmh, right. But I can promise you it is not to edify readership. It's just that there's not to much to find on uniform spaces as a general concept, so I found it reasonable to record it on MSE. There are also MSE meta discussions on what seems accepted on MSE. – freishahiri Dec 21 '14 at 04:20
  • @PeteL.Clark: Please have a look at: http://meta.math.stackexchange.com/questions/4680/is-answering-own-question-okay – freishahiri Dec 21 '14 at 04:21
  • "It's just that there's not to much to find on uniform spaces as a general concept" This material is treated in many textbooks. If you are unfamiliar with them, perhaps you'd like to ask to ask a question about that. – Pete L. Clark Dec 21 '14 at 04:27
  • @PeteL.Clark: Aah, maybe I was wrong. My excuses! – freishahiri Dec 21 '14 at 04:31
  • 1
    Also, if it's true that your sole purpose of asking the question is to record material on the site, shouldn't you delete the question upon learning that the material already appears on the site in a more general form? – Pete L. Clark Dec 21 '14 at 04:33
  • @PeteL.Clark: The reference above hides the whole subject in metric spaces and in this concern I think it is reasonable to keep this thread open, IMO. – freishahiri Dec 21 '14 at 04:39
  • @PeteL.Clark: Btw, thanks that you're willing to talk about it. I respect that! :) – freishahiri Dec 21 '14 at 04:42

1 Answers1

1

Yes it can happen!

Given the real line $\mathbb{R}$.

Consider the metrics $d(x,y):=|x-y|$ and $d(x,y):=|\arctan x-\arctan y|$.

So both give rise to the same topology.

But they cannot be equivalent as: $x_n:=n:\quad d(x_m,x_n)'\to0$

(Note how sublteties arise on the uniqueness of finite dimensional TVS.)

(Caution also that it reveals incompatibility with uniform structure of TVS.)

freishahiri
  • 17,045