In a quantum gravity discussion ("Chunkymorphism" thread) some issues of basic topology and measure theory came up. Might be fun to have a thread for such discussions.(adsbygoogle = window.adsbygoogle || []).push({});

for instance the statement was made, apparently concerning the real line (or perhaps more generally) that a countable set must consist entirely of isolated points

this is a purely topological matter, separate from measure theory (although it came up when the talk was also about some basic measure theory issues as well)

At least two of us IIRC made the point that this is mistaken, a countable subset of the reals may have no isolated points at all. Indeed the rational numbers are an example: they are countable and have no isolated points.

This brings up basic definitions in topology, like what is an isolated point, about which we could have a refresher thread if people want.

It seemed to me after I (for one) had mentioned the rationals and said you could have a countable set with no isolated points, that I was being asked "well, what does that have to do with measure theory? nothing, right?"

Right. It's purely a pointset topology thing.

However some measure theory questions came up at the same time.

Very often in measure theory you get sets of measure zero arising as exceptional sets and some statements were made or implied about sets of measure zero. It wasnt always clear what was being proposed but IIRC statements like

sets of measure zero cannot be connected

sets of measure zero can be at most countably infinite

sets of measure zero consist of isolated points

It might be fun to consider statements like these: either to find counterexamples so as to see why they arent true in general or perhaps to

figure out in which special circumstances they might be true!

there are some more or less standard measures on R^{n}which

people usually assume are meant, nothing said to the contrary, but

one is free to invent unusual measures---can you think of a topology on the reals and a measure such that any set of measure zero must consist solely

of isolated points?

Another basic question, a kind of beginning exercise, might be to prove that the rationals do in fact have measure zero----with an ordinary measure on the real line.

I was hoping someone else would start this thread. It is a utility for discussing basic topology and measure theory if people want. And if they dont thats fine with me!

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# Miscellaneous pointset topology and measure theory

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