Assume a spherical rabbit

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Hatrack River Forum: Assume a spherical rabbit (musings on topology and topologists): "t"

This person has a knack of explaining math very well without being condescending. You'll find the Poincaré conjecture, Riemann hypothesis, Nash embedding theorem etc. all explained. And unlike certain New York Times writers, s/he does not say wrong things in the interest of "reaching out to the lay person".

Or consider Einstein's "explanation" of relativity.

When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it's longer than any hour. That's relativity.

What a fucker!! I'm sure the lay person left with a very enhanced picture of relativity in his/her head. That sounds to me like Einstein saying "You're not going to understand it anyway. So let me give you some bullcrap instead. If you are smart, you'll be insulted that I chose to do this. But most probably you'll just tell people what a down-to-earth person I am"

Enjoy your visit to these sites! Thanks for finding the first link, B.

The Whitney and Nash embedding theorems

Here's a blog with a few words about the Whitney and Nash embedding theorems and some other interesting things.

David Guarrera's Worldsheet: The Levi-Civita Connection and Torsion Gravity

Generic in the Baire sense and true almost everywhere

I was talking to a friend (you know who you are) about this the other day and thought I might as well put it up here if anyone finds this useful. You might often here someone saying that some property is generic or that it holds almost everywhere in some space X. Example, "the generic continuous bounded functional on [0,1] is not differentiable anywhere!".

Genericity in the Baire sense is a purely topological concept. First, a couple of definitions:

1. A set B is called a nowhere dense set if the closure of B has empty interior.
2. A countable union of nowhere dense sets is called a meager set.

One can see that a nowhere dense set is in some sense a lightweight and since we allow only finite or countably infinite unions, a meager set too is pretty insubstantial.
3. "A property is generically true on X" means that the set where the property is false is only a meager set.

Now for "almost everywhere".
4. A statement is true almost everywhere in a measure space X if it is false only on a set of measure zero.

(3) is a statement about a topological space and (4) is about a measure space. So neither statement implies the other. Sometimes they do coincide. Consider the real number line with the usual topology and the Lebesgue measure. Then "the generic real number is irrational" and "almost every real number is irrational" are both true. Even more trivially "the generic real number is not zero" and "almost every real number is non-zero" are both true. You will see that this is because the Borel sigma algebra that the Lebesgue measure is defined on, is generated by the usual topology.

But consider the following examples.
5. Consider the interval [0,1] with the usual topology and the Lebesgue measure. Consider fat Cantor sets. They can be made to have measure as close to 1 as one wishes. However they are meager.
6. Consider the point mass measure focused at zero on the real line and the usual topology. Then the interval [1,2] would have zero measure but would certainly not be meager.

Thus neither "zero measure" nor "meager" implies the other.