Please correct me if I am wrong:
We need the general notion of metric spaces in order to cover convergence in $\mathbb{R}^n$ and other spaces. But why do we need topological spaces? What is it we cannot do in metric spaces?
I have read the answers at Motivation of generalizing the theory of metric spaces to the theory of topological spaces and want to emphasize this example I found in Preuss "Foundations of Topology":
Sorry for the big image, but I want to be sure you know what I mean. So does this mean we cannot describe pointwise convergence in a metric space? Can you elaborate more on this specific example? I don't really see the conclusion.
Another point is that Preuss explains that continuous convergence cannot be described in topological spaces (I am not sure if he is referring just to Hausdorff-spaces here).
Best Answer
The point Preuss makes is that we cannot find a metric $d$ on the set of function such that "$f_n \to f$ pointwise" is equivalent to "$f_n \to f$ in the metric $d$" or $d(f_n,f) \to 0$ etc. Uniform convergence does correspond to a metric (from the supremum-norm). But we can define something more general, a topology, such that we can define $f_n \to f$ in that topology , and moreover in such a way that it exactly corresponds with "$f_n \to f$ pointwise".
The following deficiency of topologies is that a metric defines a topology (but not always conversely) but whereas in metric topologies sequences actually suffice to completely describe that topology, in general topologies this is no longer the case and the familiar (from analysis/calculus) sequence must be replaced by a more general notion of convergence, sequential continuity does no longer suffice (we need general continuity), sequentially compact has to be replaced by general compactness; all of these are mostly improvements (in the sense that the general properties behave better wrt topological constructions), but less familiar (in topology the idea to call a set "compact(like)" iff every sequence has a convergent subsequence, comes from analysis and is older (and often more directly applicable too)).