fourier jr said:
more like analysis is the study of limits. i don't see the word limit anywhere in any toplogy text, not even in the stuff about nets & filters, which are generalizations of the sequence concept. to have limits you need a metric & to have a metric you need a metrizable space, but not every space is metrizable. the topological definition of continuity also doesn't use any limits; it says that the preimage of an open neighbourhood of f(x_0) in the target space is an open neighbourhood of x_0 & that's all.
No, analysis includes a lot more than limits. Analysis is basically the theory behind calculus- and, as I said before, you need at least a topological vector space for that.
I certainly have seen limits in topology texts: Suppose {a
n} is an infinite sequence of points in a topological space. Then a
n-> n (as n goes to infinity) if and only if, for any open set containing a, there exist N such that if n> N then a
n is also in that open set.
Same thing with limits of functions. If f: M-> N is a function from topological space M to topological space N, x
0 is in M, then limit f(x)= b (as x->x
0), b in N, if and only if for every open set V containing b, there exist an open set U containing x
0 such that f(U) is a subset of V.
That, of course, would give the same definition of "continuous function" as the one you give.