A family $\mathcal B\subset\mathcal U$ is called a base for the uniformity $\mathcal U$ if for every $V\in\mathcal U$ there exists a $W\in\mathcal B$ such that $W\subset V$. The smallest cardinal number of the form $|\mathcal B|$ where $\mathcal B$ is a base for $\mathcal U$, is called the weight of the uniformity $\mathcal U$ and is denoted by $w(\mathcal U)$.
- Can the weight of the uniformity be expressed using cardinal invariants of the topology $\mathcal T$ induced by the uniformity $\mathcal U$.
For example, it is clear that if $\{B[x]; B\in\mathcal B\}$ is a local base at the point $x$. Therefore $\chi(\mathcal T)\le w(\mathcal U)$, where $\chi(\mathcal T)$ denotes the character of the topological space $(X,\mathcal T)$.
- Is it true that $\chi(\mathcal T)=w(\mathcal U)$?
(I believe that the weight of the uniformity is usually denoted by $u(X)$ or $uw(X)$ to avoid confusion with $w(X)$, the minimum cardinality of a base for the topology on $X$.) A Tikhonov space with countable uniform weight is metrizable. Every Tikhonov space is uniformizable, so just take a non-metrizable, first countable Tikhonov space, like $\omega_1$ with the order topology, and you’ll have a space whose character is less than its uniform weight.
I’ve never done much of anything with uniform spaces; the only non-obvious result that I know relating $u(X)$ to other cardinal functions is that $u(X)\le w(X)\le u(X)c(X)$, where $c(X)$ is the cellularity.