Surely, continuity is a very basic topological concept. But I don’t recall any explicit definition of limit of a function as we find in calculus.

I think it is incorrect to mark this question as a duplicate. The cited other question asks for sources of where limit has been defined in topology. I ask why limit of a function is not usually defined. Indeed, Noah’s answer addresses this question.

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    $\begingroup$ That's because limits in the usual $\epsilon-\delta$ definition require the notion of distance, which is lacking in a general topological context. Instead, continuity is defined in terms of open sets, i.e. directly in terms of the topology. The two definitions coincide when a topological space does have a notion of distance, i.e. it is a metric space. $\endgroup$ – NickD Feb 21 at 15:32
  • $\begingroup$ Seems to me you could define limit generally as follows. Let X and Y be topological spaces, a in X, A a subset of X, f a function from A to Y, and c in Y. c is a limit of f as x approaches a iff for any neighborhood N of c, [f^-1](N) u {a} is a neighborhood of a. (My neighborhoods need not be open; rather, the point must be in the interior.) $\endgroup$ – Stephen Herschkorn Feb 21 at 15:56
  • $\begingroup$ Calum- Only a little. $\endgroup$ – Stephen Herschkorn Feb 21 at 18:25
  • $\begingroup$ Nick- The whole raison d’être of general topology is to develop a notion of “arbitrarily close” without reference to distance or metric. $\endgroup$ – Stephen Herschkorn Feb 21 at 20:09
  • $\begingroup$ You could, using net convergence, define a limit $\lim_{x \to a} f(x)$ as follows, if $a$ is not an isolated point of the domain: if there is some $b \in Y$ such that whatever net $(x_i)_{i \in I}$ in $X\setminus \{a\}$ we take with $\lim_i x_i = a$ then we always have $\lim_i f(x_i)=b$, then we call $b$ a limit of the function as $x$ tends to $a$. This need not be unique. It's quite hard to check in practice as well. $\endgroup$ – Henno Brandsma Feb 21 at 22:41

The usual notion of limit generalizes naturally to arbitrary topological spaces: given spaces $(X,\tau)$ and $(Y,\sigma)$ and a map $f:X\rightarrow Y$, we can say $\lim_{x\rightarrow a}f(x)=b$ (with respect to $\tau$ and $\sigma$) iff $$\forall U\in\sigma\exists V\in\tau(b\in U\implies V\not=\emptyset\wedge\forall x\in V\setminus\{a\}(f(x)\in U)),$$ or in words iff for every open set $U$ in the target containing $b$ there is some open set $V$ in the source containing $a$ which $f$ sends entirely inside $U$ (with the possible exception of $a$ itself). It's not hard to check that this agrees with the usual notion of continuity in analysis, and for reasonably nice spaces it's single-valued so actually makes good sense (to see the potential issue, consider the indiscrete topology; it's a good exercise to pin down what "reasonably nice" means in this context).

So why don't limits appear more centrally in a basic point-set topology class (at least in the context of spaces in which they are guaranteed to be unique when they exist)?

Well, first of all they do, at least partly: limits of sequences are in fact things we care about rather frequently. It is true, though, that we talk much less (at least initially) about limits at specific points of functions between general topological spaces, focusing more on the question of overall continuity. So the question still remains.

I think the situation is best explained by recalling why we care about limits in classical analysis in the first place: we're trying to give a formal treatment of calculus, and the key notions of calculus - derivatives and integrals - both correspond to taking limits of appropriate functions at appropriate points. But the apparatus of calculus does not generalize well to arbitrary topological spaces, and so local questions like "What, if anything, is $\lim_{x\rightarrow a}f(x)$?" wind up being less important (at least initially) than global questions like "Is the map $f$ continuous?" The limit idea becomes primarily a tool for answering such questions - e.g. to show that a function isn't continuous we find some point where its limit either doesn't exist or differs from its actual value. But we can develop such arguments without using the limit language.

Of course that's a subjective explanation which many might reasonably disagree with, but I think it's more-or-less accurate. That said, I do think that introductory topology classes should indeed present this generalization if only as an example of how to use topological ideas (I did when I taught it, and if I recall correctly Munkres' book does as well).

  • $\begingroup$ I would make f a partial function, since that is often the applications. See my comment to the question. $\endgroup$ – Stephen Herschkorn Feb 21 at 18:26
  • $\begingroup$ @StephenHerschkorn Any partial function is total when we restrict to its domain as a subspace of the original space - since the shift to subspaces is something we do all the time in topology, I'm fine with doing that here. But certainly there are many ways to tweak the above. $\endgroup$ – Noah Schweber Feb 21 at 18:51
  • $\begingroup$ The point is we often take the limit at points outside of the domain. $\endgroup$ – Stephen Herschkorn Feb 21 at 20:05
  • $\begingroup$ @CalumGilhooley I believe this is visible to all users? $\endgroup$ – Noah Schweber Feb 25 at 15:45
  • $\begingroup$ @CalumGilhooley I think it is incorrect to mark this question as a duplicate. See the edit of the question. Is there a way to undo a closure? $\endgroup$ – Stephen Herschkorn Feb 25 at 20:49

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