Generates a covering uniformity on a set X, by an entourage uniformity.

Question

A family $\mu$ of coverings of a set $X$ is called a covering uniformity if it satisfies the following conditions:

1. If $\frak U,\frak V\in\mu$ there is a $\frak W\in\mu$ with $\frak W\prec \frak U$ and $\frak W\prec \frak V$,
2. every element of $\mu$ has a star refinement in $\mu$,
3. if $\frak U\in\mu$ and $\frak U\prec \frak W$ then $\frak W\in\mu$.

Let us note that, A basis for a covering uniformity is a family of coverings that satisfies conditions (1) and (2).

Now If $\mathcal U$ is an entourage uniformity for $X$, for each $U\in\mathcal U$, let $\mathfrak{T}_U= \{ U[x]:x\in X \}$ and put $\mu=\{\mathfrak{T}_U:U\in\mathcal U\}$.

then $\mu$ is a covering uniformity or a basis for a covering uniformity? i can show that $\mu$ satisfies conditions (1) and (2).

Answer 1

These uniform covers $\mathfrak{T}_U$ only form a base for the covering uniformity.

Quote from Willard, General Topology (p. 245)

Although, as we have said, coverings and surroundings should be used in the same way as one uses open and closed sets in a topological space, i.e. interchangeably, we should comment that the passage back and forth is not nearly as neat. The uniform covers of a uniform space translate only to a base for the surroundings and similarly, the surroundings provide, in translation, only a base (as defined below) for the uniform covers. This causes no real problems, since all the important concepts defined for uniform spaces can be defined in terms of bases for the uniformities in question.

The usual approach is to take a base for the entourage uniformity and the corresponding uniform covers form a base for a covering uniformity, and vice versa, if we have a covering uniformity with base $\{\mathfrak{U}_i : i \in I\}$ then the entourages $U_i = \cup \{ U \times U : U \in \mathfrak{U}_i\}$ for a base for an entourage uniformity. These uniformities generate the same topology etc., and are equivalent (also in uniform sense). See exercise 36C in Willard.