lugita15 said:
Why is it that continuous functions map convergent sequences to convergent sequences, but not Cauchy sequences to Cauchy sequences? That's somewhat counterintuitive.
WbN's link is pretty neat. But allow me to give a bit more of a higher level answer.
First, let me look at metric spaces. Take two points ##x## and ##y## in a metric space. We say that they are ##\varepsilon##-close if ##d(x,y)<\varepsilon##. So metric spaces allow us to measure when two points are sufficiently close to each other.
Of course, this definition uses the metric in a very crucial way. However, the entire point of a topology is to be able to generalize the above. Indeed, in a topological space, we can take a neighborhood ##U## of a point ##x##. We then say that ##y## is ##U##-close to ##x## if ##y\in U##.
So a topological space is able to say when a point ##y## is very close to a point ##x##. Indeed, we can say that ##y## is "very close" to a point ##x## if ##y## is ##U##-close to ##x## for "many neighborhoods" of ##x##. This is all just intuitive of course.
However, this makes things clear that we can describe convergence of sequences in a topological space. Indeed, let ##(x_n)_n## be a sequence. We say that ##x_n\rightarrow x## if for any neighborhood ##U##, there exists a point ##m## such that ##x_n## is ##U##-close to ##x## for ##n\geq m##.
The crucial thing to notice here is that we compare the sequence elements to a fixed point ##x##. We don't let ##x## vary. The same with continuity. We define continuity at a point ##x##. The crucial point is again that we don't let ##x## vary.
However, when we look at Cauchy sequences, we see that something different is going on. In the definition of a Cauchy sequence, we see that for all ##p,q>N## we have that ##d(x_p,x_q)## are ##\varepsilon##-close. So here, the ##x_p## and ##x_q## vary. It is not clear (or possible) to write this definition using some notion of ##U##-closeness.
The problem is this. Given ##4## ponts ##x##, ##y##, ##z## and ##t##. In a metric space, we can easily say that ##x## and ##y## are as close to each other as ##z## and ##t## are. Indeed, we just compare ##d(x,y)## and ##d(z,t)##.
In a topological space, we can say that ##y## is ##U##-close to ##x##. But ##U## is a neighborhood of ##x##. It has nothing to do with ##z## and ##t##. So we can't say that ##z## is ##U##-close to ##t##, since ##U## isn't a neighborhood of ##t##. Neither is there any way to "translate" a neighborhood of ##x## to a neighborhood of ##y##. So the intuitive problem here is that neighborhoods of one point don't have anything to do with neighborhoods of another point.
A space where we make this possible is a uniform space. A uniform space is a set ##X## together with a uniformity. A uniformity is a collection of subsets of ##X\times X## that satisfies some axioms. Take an element of the uniformity ##U##. Now we can say that ##x## and ##y## are ##U##-close if ##(x,y)\in U##. Take our four points from above. We can say that ##(x,y)\in U## and ##(z,t)\in U##. This would measure that ##x## and ##y## are as ##U##-close to each other a ##z## and ##t##.
Now we can make sense of the notion of a Cauchy sequence. Indeed, we say that ##(x_n)_n## is Cauchy if for all elements of the uniformity ##U## holds that there exists a ##N## such that for all ##p,q>N## holds that ##x_p## and ##x_q## are ##U##-close.
Clearly, a metric space induces a uniform space. We take as uniformity the sets ##\{(x,y)~\vert~d(x,y)<\varepsilon\}## (and some combinations of those for technical purposes).
Every uniform space also induces a topological space. Indeed, given a point ##x## and an element of the uniformity ##U##, we say that ##\{y\in X~\vert~y~\text{is U-close to x}\}## is a neighborhood of ##x##. This determines a topological space.
To make a long story short. Cauchy sequences are no topological concept. They are a uniform concept.
