# Alternative to epsilon-delta definition

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?

• He probably had the definition of continuity between topological spaces in mind. – Git Gud Jun 11 '14 at 15:18
• @GitGud: Agreed. Once you go pullback, you never go ... hmm... – Eric Towers Jun 11 '14 at 15:20
• I think by "don't use the epsilon delta defintion" your prof probably actually means "use a more general definition than the epsilon delta definition." Teachers use the epsilon-delta definition because it decreases the abstraction load on the student, and models the relationships more concretely. Of course, eventually we hope we can get you comfortable to make the jump to abstraction :) – rschwieb Jun 11 '14 at 15:38

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.

• What is 'N'? -- Other than a sufficiently large number? – user156541 Jun 11 '14 at 15:42
• Just some large enough integer. It's saying that all except for finitely many of the $x_n$ are in $U$. – user98602 Jun 11 '14 at 15:46
• I haven't studied topology, yet -- and probably not enough about metric spaces. Would Munkres' book cover this? – user156541 Jun 11 '14 at 15:48
• Absolutely!${}$ – user98602 Jun 11 '14 at 15:51
• Good. I'll be spending some time with Munkres' Topolgy later this year. Thanks. – user156541 Jun 11 '14 at 15:53

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, $$\forall S\subset F \,\,\, S \text{ is open and }x\in S \implies x_n\in S \text{ for }n\text{ big enough}$$

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

• I haven't looked at his fourth edition, but on pg 84 of the third edition is the epsilon-delta definition. I haven't read that book in quite a while, but I don't remember (and skimming through, could not find) another definition (other than the "provisional definition" a few pages before). – user156541 Jun 11 '14 at 15:28