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I'm using the uniform cover definition of a uniform space.

Let $f:(X, \mu)\to (Y, \nu)$ be a surjective map such that $(X, \mu)$ is a uniform space and $\nu$ is a largest preuniformity on $Y$ such that if $\mathcal{U}\in\nu$ then $f^{-1}(\mathcal{U})\in\mu$.

I want to prove that then $\nu$ is a uniformity. I've proved that if $\mathcal{V}\in \mu$ then $f(\mathcal{V})\in \nu$ (this is a cover since $f$ is surjective).

I've actually proved this hoping that a cover which separates some of the fibers in $X$ will arise and I'd be able to use it to separate two points in $Y$. I'm stuck.

Jakobian
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  • are you using entourages or covers? What is the difference between a preuniformity and a uniformity? – Henno Brandsma Jul 19 '21 at 21:19
  • @HennoBrandsma uniform covers. A uniformity seperates points - that is, for any two points $x, y$ there is a uniform cover $\mathcal{U}$ such that any $U\in\mathcal{U}$, we can have $x\in U$ or $y\in U$, but not both. A preuniformity doesn't have to. A uniform space is then a Tychonoff space while a preuniformity doesn't have to be. – Jakobian Jul 19 '21 at 21:24

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