# Extending a function beyond the completion/closure of its domain

In analysis there are certain theorems that tell under which conditions you can continuously extend a continuous functions to the closure/completion of its domain (which actually give the same set, since when talking about completions we have to be (at least) in a metric space).
The theorems I have in mind are these three:

• Every uniformly continuous mapping between a metric space and a complete metric can be extended uniformly continuous to a mapping between the completion of the domain and the range.
• Every continuous function on a dense subset can be continuously extend to the closure of that subspace (which is by definition of "dense", the whole space).
• The theorem from here.

What I'm looking for is a collection of theorems that tell me under which circumstances one can extend a continuous function beyond the completion or the closure of it's domain.
The only theorem of this kind that I know of is the famous Tietze-Urysohn extension theorem (which applies to function whose domain is already a closed set).

I'd be also happy with a reference, as long as the theorems listed there are quickly accessible (i.e. don't require reading through a thicket of abstract/very specialized definitions before getting to the theorems - the more "concrete" the theorems are (as in $\mathbb{R}^n$, or a metric space), the better.

The recent two-volume book Methods of geometric analysis in extension and trace problems by Brudnyi and Brudnyi looks like the reference you want. It has separate chapters for results for $\mathbb R^n$ and for general metric spaces.