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I'm currently doing an undergraduate course on topology and am currently studying uniformities. I suppose this question is very simple but I just am completely stuck.

I understand how to, having the base of a covering uniformity, construct the base for a diagonal one. I also understand the opposite direction.

If one starts with a diagonal uniformity, constructs a covering uniformity from it, and then constructs another diagonal uniformity from this last one, one ends up with the original diagonal uniformity. The same applies if one starts with a covering uniformity.

Quoting Willard:

  1. Let $\mu'$ be a base for a covering uniformity $\mu$ on X. Then the collection of all sets $D_u = \bigcup \{U \times U | U \in u \}$, for $u \in \mu'$, is a base for a diagonal uniformity on $X$ whose uniform covers are precisely the elements of $\mu$.

  2. Let $d'$ be a base for a diagonal uniformity $d$ on $X$. Then the collection of all covers $\mathcal{U}_D = \{D[x] | x \in X\}$, for $D \in d'$ , is a base for a covering uniformity $\mu$ on $X$ whose surroundings are precisely the elements of $d$.

However, let's say that we start with a covering uniformity. Our teacher has asked us to prove that if we take an individual cover from it and apply the transformation process above, we don't get the same cover back. This is what I don't understand: where is the cover lost in the process? I have the same question in the case of starting with a diagonal uniformity.

Thank you for your time!

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I have now understood we don’t get the same cover because of the overlapping of the various D[x]’s! Maybe if there was a partition one would get the same one.

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