« first day (1130 days earlier)      last day (3336 days later) » 

Martin Sleziak
Martin Sleziak
11:10 AM
20
Q: Continuity and the Axiom of Choice

Donkey_2009In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies $f(z_n) \to f(a)$. $(2)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if $\forall ...

analysis set-theory axiom-of-choice
I like the fact that:
a) We need (some form of) AC to prove that the two definitions of continuity at a point are equivalent.
b) If we consider global continuity, then the two definition are equivalent in ZF. Although the proof is more complicated. (This result is due to Sierpniski.)
 

« first day (1130 days earlier)      last day (3336 days later) »