I am trying to learn a bit about linear topological rings and modules. In doing that, I have stumbled upon some questions about uniform spaces and I am struggling to find references. My main is question is as follows:
Suppose that $X$ is a uniform space. Then one can take its Hausdorff completion $\hat{X}$: this is a uniform space that is complete and Hausdorff. There is a canonical map $t:X \to \hat{X}$ which is uniformly continuous and is characterised by the following universal property: every uniformly continuous map from $X$ to a uniform space which is complete and Hausdorff factors through $t$ via a unique uniformly continuous map. Is it true that $X$ being complete implies that $t$ is a surjection? And what about the converse?
Any guidance or standard references would be extremely welcome.