I am trying to exactly understand what is meant by the term "technical machinery". Particularly I was reading the history about compactness and it came up in this passage:
"It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space."
So what does "technical machinery" really mean?
Expanding my earlier comment into an answer:
It's an informal term, and as such you're not going to find a precise meaning. It refers to any additional work (in the form of results to be proved or auxiliary concepts to be made precise) needed to set up a proof or a definition. Besides that, the phrase generally indicates:
An emphasis on the technical aspect - the phrase "technical machinery" is generally used to refer to the "nuts and bolts" of such set-up, and not so much the intuition behind that additional set-up.
A sense of "self-sufficiency" of this set-up - the technical machinery of FOOs isn't just something you use to prove the BLAH conjecture, it's (potentially) useful for other problems too or it's interesting in its own right.