Convergence and topology , Intuition

Question

I am familiar with the definition of topology on sets, (i.e. a structure closed under union and finite intersection). However, when it comes to statements like given two spaces the convergence , e.g. that $f_n\to f$, introduce a topology. I have problem to relate these two concepts together. Moreover, also does any mode of convergence would introduce the same intuitive topological result?

Answer 1

The elements of the topological structure are regarded as the open sets. A closed sets is defined as the complement of an open set. (Thus these are closed under finite union and arbitrary intersection.)

In a topological space a sequence $x_n$ is said to converge to the ('limit') point $x$, if every open neighborhood $U$ of $x$ (i.e. open set containing $x$) contains all but finite elements of $\{x_n\}$, that is, there is an $N$ such that $x_n\in U$ if $n>N$.

We can prove that any subsequnce of a convergent sequence converges to the same limit point, and that if a closed set contains a converge sequence, it also contains its limit point.

---

If we are given a set $X$ and certain sequences of $X$ together with their 'wannabe-limit-points', then this data induces a topology on $X$: namely we define a set $S\subseteq X$ closed if whenever $S$ contains an infinite subsequnce of a given sequence, it also contains its 'wannabe-limit-point'.

Now in this topology, any given sequence $x_n$ indeed converges to its given limit point $x$, as if $x\in U$ with an open $U$, we can't have infinitely many $x_n$'s in the closed $X\setminus U$, since then it would define a subsequence of $x_n$ and $x\in X\setminus U$ would follow.

Different convergence data, of course, usually induce different topologies.