Let me apologize straight away for my prolixity - I have been playing village cricket this afternoon which inevitably involves a few beers after. Also, Mrs. G. is at a concert tonight, so I am slightly bored.

Right. It is no secret that I dislike pure analysis - as a hobby mathematician I am free to pick-and-choose what I study, though I would be the first to admit that neglecting the "fundamentals" may well be unwise. This neglect has tripped me up more than once. Ho hum.

For reasons nobody here needs to know, I decided to study functional analysis, and I work from a text by Walter Rudin, again admitting I have a weak grounding in pure analysis.

Anyway, on first reading Rudin's opening chapter (intro to topological vector spaces etc) I am arrogantly thinking, like, yeah, yeah, I know all this stuff and consequently skimmed. But then a coupla chapters down the line I got confused by notation and terminology, so went back to the Intro.

Which brings me finally, and somewhat boringly, to my question, and I use Rudin's notation exactly.

Suppose that is a vector space and that . Then is said to beconvexif for all . Note that this is shorthand for the following:

Whenever then for all . The intuitive meaning is clear, provided only that we are prepared to bend the "rules" more than a little. It says that the "further you move incrementally from "x" the closer you get to "y" and still stay in the set".

Yeah, I know, you're right to say "Ugh"!

So Rudin now defines a topology on in the obvious way.This seems to suggest to my pea brain that all the above can be written as follows:

Define a curve in by such that . Then if, for all that for any

defines a path-connected topological space.

Or have I taken a wrong turn early on?