Topological Spaces: Pre-Uniform Structures

Question

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!

Answer 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.)