Alternative to epsilon-delta definition

Question

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go over it in more depth in real analysis, and those that wouldn't continue didn't need to know it, anyway), said that modern mathematicians don't use the epsilon-delta definition that was in the book, anyway. He alluded to some kind of set definition. I wasn't particularly interested at the time, but now I'm wondering what he was referring to.
A cursory (very cursory, in fact) search of the internet did not yield said definition. So I thought I'd just ask. Do you guys know what definition of limits he was talking about?

Answer 1

He's likely referring to the notion of a continuous map of topological spaces. A topological space is a set equipped with a set of "open sets" satisfying certain properties; familiar examples are metric spaces, for instance.

A map $f: X \rightarrow Y$ between topological spaces is said to be continuous if, for every open set $U \subset Y$, $f^{-1}(U)$ is open in $X$. To motivate this, you might want to prove that this definition is equivalent to the $\epsilon-\delta$ one for metric spaces.

In a topological space, one can say that a sequence $(x_n)$ has limit $x$ if, for every open set $U$ that contains $x$, $x_n \in U$ for $n > N$ (with $N$ sufficiently large). (Check that for metric spaces, this is equivalent to the standard definition!) This definition can be turned into one of 'continuity' as above. In general, the notion of limit of a sequence isn't well-behaved in a topological space; one needs to work more abstractly with the notion of net, which is baggage you shouldn't worry about right now. In a Hausdorff space, they behave (mostly) as they should: a sequence has at most one limit.

Answer 2

If you have this book available to you, I believe it uses the definition you're looking for. It's also a great foundational book if you're thinking about getting waaay deep in math. http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

Answer 3

Define open sets as sets $S$ for which $$ \forall x\in S\,\,\, \exists r>0\,\,\, \forall y\,\,\, |x-y|<r\implies y\in S $$ Then the definition becomes: $x_n \in F\to x\in F$ if, $$ \forall S\subset F \,\,\, S \text{ is open and }x\in S \implies x_n\in S \text{ for }n\text{ big enough} $$