Is continuity of topological spaces appears only in the context of normed topological spaces?

Question

Thinking about continuity in the real line which makes use of epsilon and delta neighbourhoods (delta and epsilon being metrics), I suppose that there has to be some notion of distance used in continuity in general topological space. I only have a degree in electrical engineering and being interested in the applications of functional analysis.

Answer 1

Not at all!

Part of the motivation for topology in the first place is the realization that concepts normally associated with distance - e.g. continuity, connectedness, and compactness - can actually be characterized using only the notion of openness. For example:

• A function $f:X\rightarrow Y$ between topological spaces is continuous iff for each open $U\subseteq Y$ the preimage $f^{-1}(U)$ is open in $X$.

• A topological space $X$ is connected iff there is no pair of disjoint nonempty open sets $U,V\subseteq X$ such that $U\cup V=X$.

• A subset $C$ of a topological space $X$ is compact iff for every collection of open-in-$X$ sets $\{U_i\}_{i\in I}$ such that $\bigcup_{i\in I}U_i\supseteq C$, there is some finite $J\subseteq I$ such that $\bigcup_{i\in J}U_i\supseteq C$.

(OK fine, I'm being a bit sloppy here in conflating a topological space with its set of points - there are many ways to put a topology on a given set of points, as long as that set has more than one element - but this is a minor issue.)

These are "faithful rephrasings" of the original metric notions in the following sense: to each metric space $(X,d)$ there is a corresponding topology $\tau_d$ on $X$, and when we translate from $d$s to $\tau_d$s the metric notions correspond to the versions above. Meanwhile, since there are lots of topological spaces not coming from metrics at all, we get a real gain in generality.

So no metric structure is needed at all to make sense of the standard "continuity-flavored" ideas in real or functional analysis. This isn't to say that such additional structure (beyond a mere topology) isn't useful or interesting, but it's not necessary for the expression and basic analysis of many simple properties.

At this point there's a reasonable follow-up question:

How different can topological spaces really be from metric spaces - especially if we restrict attention to reasonably natural spaces, or spaces of interest outside pure topology?

The utterly amazing book Counterexamples in topology addresses the looser question wonderfully, and examples of more natural/broadly interesting spaces which are extremely different from metrizable spaces can be found in various places on this site and mathoverflow - e.g. here or (even more extremely!) here. Given your background, it's worth mentioning that several examples at each of those links are from functional analysis. The takeaway should be that the "topological shift" really is valuable and significant.