6 Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets. An immediate consequence is that the $\sigma$-algebra is uncountable.

0 There is a way of proving that there is "infinite number of sides in a circle", in the following sense. Infinity was a number to both Kepler and Leibniz who spoke of a circle as an infinite …

Are you familiar with Taylor series? Series solutions of differential equations at regular points? From what foundation/background are you approaching this problem?

What do finite, infinite, countable, not countable, countably infinite mean? [duplicate] Ask Question Asked 13 years, 2 months ago Modified 13 years, 2 months ago

Just out of curiosity, I was wondering if anybody knows any methods (other than the integral test) of proving the infinite series where the nth term is given by $\frac {1} {n^2}$ converges.

A set X is countably infinite if there exists a bijection between X and $\mathbb {Z}$. To prove a set is countably infinite, you only need to show that this definition is satisfied, i.e. you need to …

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining …

All three integrals are divergent and infinite and have the regularized value zero, but two of them are equal but not equal to the third one. In other cases of divergent integrals or series, the …

Why is the infinite sphere contractible? I know a proof from Hatcher p. 88, but I don't understand how this is possible. I really understand the statement and the proof, but in my imagination this...

Richard Thompson's groups $T$ and $V$ are well-known examples of infinite simple groups. See this answer of mine for more details, or look up the article Introductory notes on Richard …