Why is the topological definition of continuous in terms of open sets?
I think my main complaint might be that the notion of open set seems too flexible/general and considers too many things that don't seem the right notion of "closeness". Conceptually, people explain "continuous" as:
Nearby points map to nearby points.
But we can easily construct sets for which *all their points are not “nearby” but they are still open. A simple example in metric spaces: the union of two open balls. The sets are still open but the points in one ball vs the other are not nearby. However the topological definition is in terms of open sets so it would consider maps balls like this from $X$ to $Y$ while that doesn't seem right to me. Is there something that I am missing?
I guess I find it better to have a notion that captures the idea of “balls of radius epsilon in Y” to “balls of radius delta in X” a better notion of continuous.
Another issue I find with this is that I find this in conflict even with the traditional epsilon-delta definition. The way I see it is that the topological definition should be more general (and abstract) and should encompass the metric space definition as a special case. Which to me it’s not clear it does because there is this union of disjoint open sets issue, that seem get included in the topological definition but for me they shouldn’t. This point seems important. Why were open sets chosen as the correct notion? A better definition for me would be (instead of open sets) to be in terms of “balls of radius epsilon in Y” to “balls of radius delta in X” in some topological way to define this.
I have of course read the descriptions of open sets in wikiepdia but that doesn't seem to really clarify things. I know that open sets are the set of points under some topology that are "close". i.e. we only need sets to classify what points are considered "close". Which seems to me the main motivation why open sets were chosen, but the fact that disjoint open balls pass the test and are considered "close by" particularly disturbs me for some reason. Why is this specific complaint OK to ignore? What justifies not being worried about it?
Another reason I find it weird to use open sets is because for me open sets (since I am most familiar with the definition of open sets in metric spaces), are a type of set where everything is an interior point. It's a type of set that:
for all points we can always find a perturbation such that the point remains in the set (thus there is a neighbourhood that contains it in E).
I find this problematic since it doesn't seem the right notion of "nearby" (at least to me); the reasons I prefer the definition to be restricted to only single open balls or sets that have no weird gaps (continuous sets? for some definition of that). This interior point issue doesn't seem to be what continuity (or limits actually) encompass conceptually. Continuity/limits seem to be a property about getting closer and closer (at least conceptually) or approaching. Therefore, for me it would be better to define it in terms of sets that reflected this idea of closeness. Something like neighbourhoods or (open) balls like in the traditional way of defining balls $B_{\delta}(p) = { x \in X | d(x,p) < \delta}$. Since this seems to be a clear notion of "nearby". Why are these ideas not preferred? What is wrong with it?
Best Answer
I think what User Randall wrote in a comment is the main point: Only half of the emphasis in the definition of continuity as
should lie on
but at least half of it on
The intuition is that a set is open if around each point inside, there is still some wiggle room a.k.a. neighburhood around it. Granted that some open sets in a metric space also contain some points "far away", as in your example with the disjoint union of two balls -- but now that is where the all in the definition kicks in: To check continuity, you will also have to consider single balls. Really, really small single balls. All of them.
And in a metric space, it is clear that checking it for "very small" balls, small as in "your favourite $\epsilon$", suffices to prove it for all. Without a metric, it's harder to tell which open sets are small, so, well, let's just make the definition robust and demand it for all of them. (Actually, sometimes it suffices to check on various kinds of "basic" open sets.)
So the "all" is a placeholder for arbitrarily small, which technically does not make sense in a general topological space. As soon as it does -- in a metric space --, you can replace "all" by "arbitrarily small", and making that rigorous will give the usual $\epsilon-\delta$-definition back.