sniffnoy | A quick note on the notion of total boundedness, and some other abstract nonsense

It is true that in a totally bounded space every uniform cover has a finite uniform subcover (rather than just a finite subcover). But contrary to what Willard claims, this is not obvious. It is obvious once you know totally bounded => precompact (which everybody knows :P ), but it definitely should not be taken as part of the definition.

Also: The notion of uniform cover is pretty different from what I would intuitively expect that term to mean. I wonder if I can sensibly formalize the latter... (I mean, I'm pretty sure I can come up with a definition, question is whether this will actually be another way of describing the notion of uniformity, or whether it will just be terrible). More on that once I've thought about it.

Yeah, this is all pretty basic and pointless. It's bothering me anyway.

-Harry