TITLE: Why do topologists dislike sequences?
QUESTION [5 upvotes]: Okay, the header might be a slight exaggeration, but...
My introductory text on point-set topology defines convergence of point sequences in topological spaces, but then immediately states that these do in fact not play a large role in the theory of topological spaces, contrary to $\Bbb R^n$. In fact, it claims that much of the motivation behind the development of point-set topology was to get rid of the great reliance on point sequences.
But why this motivation? Why would you want to get rid of a tool as invaluable as point sequences?

REPLY [10 votes]: Somewhere in Munkres' topology textbook there is (or was in an earlier edition?) a set of exercises that prove that nets do for topological spaces generally what sequences do for metric spaces and other spaces satisfying the first axiom of countability.
Here are three familiar examples of convergence of a net:

$\lim\limits_{n\to\infty}\dfrac 1 n = 0$.
$\lim\limits_{x\to0}\dfrac{\sin x}{x}=1$.
$\displaystyle\lim_{\|P\|\to0} \sum_P f(x)\,\Delta x=\int_0^1 f(x)\,dx$ where $\|P\|\to0$ means the mesh of the partition $P$ approaches $0$ and the sum is over all intervals in the partition, and $x$ is in the interval in question, and $\Delta x$ is the length of that interval.

The third example above is the only instance of convergence of a net that is familiar to everyone who's never taken a topology course and is not either a limit of a sequence or a limit of a function of the kind treated in the usual freshman calculus course.
PS: In the exercises in Munkres, it is useful to use as the directed set whose members are the indices in the net, the set of all open neighborhoods of the point to which the net is to converge, ordered by the inverse of inclusion.