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It is surely common knowledge, more general than the $\epsilon, \delta$ definition of a limit of a function on a subset of a metric space at a limit point of the subset - see for example page 83f. of Walter Rudin, Principles of Mathematical Analysis (third edition, McGraw-Hill 1976) - that if $X$ is a topological space, $Y$ is a Hausdorff space, $E$ is a subset of $X$, $f \colon E \to Y$ is a function, $a$ is a limit point [equivalently: cluster point, accumulation point] of a subset $K$ of $E$ (this does not imply $a \in K$, or even $a \in E$), and $b$ is a point of $Y$, then a notation such as $$ \lim_{\substack{x \to a \\ x \in K}} f(x) = b, $$ or similar, means that every neighbourhood of $b$ in $Y$ contains the $f$-image of the intersection of $K$ with a punctured neighbourhood of $a$ in $X$. A recent question asked about a special case of this ($E = X = \mathbb{R}$, $K = \mathbb{R} \setminus\{a\}$, $Y = \mathbb{R} \cup \{+\infty, -\infty\}$), and I have been seeking an authoritative reference for this "common knowledge".

The only definition I have managed to find is on page 63 of Horst Schubert, Topology (Macdonald 1968). The book is sadly out of print. (Used copies of it don't seem to be very easy to find.) Also, the definition is given in terms of filters. Although not complicated, the definition requires the reader to apply quite a large number of prior definitions in order to arrive at the characterisation in terms of neighbourhoods of $b$ in $Y$ and punctured neighbourhoods of $a$ in $X$. (I quoted the necessary definitions in my answer to the question cited earlier.)

Is there a book in print that gives an explicit definition of $\lim_{x\to a} f(x) = b$ in the general case?

It would be ideal if the book gave the simple definition in terms of neighbourhoods in $Y$ and punctured neighbourhoods in $X$, but a more elaborate definition in terms of filters or nets is also acceptable. The fiddly details concerning the subsets $E$ and $K$ are relatively unimportant; what matters is that the definition applies to topological spaces in general, not just metric spaces.

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    $\begingroup$ If I recall correctly, Bourbaki's General Topology pt. I does this, although they first develop the language of filters and state it more as a remark than anything else. That's pretty authoritative I would say, and I imagine a used copy won't be that hard to find. $\endgroup$ – Guido A. Mar 13 at 17:53
  • $\begingroup$ This book, Horst Schubert's Topology is certainly reachable on the web, because I just did it (I can't provide with link though, it is probably illegal). $\endgroup$ – enedil Mar 13 at 17:54
  • $\begingroup$ @GuidoA. I don't know how on Earth I managed to miss that - very embarrassing! I checked several books, and Bourbaki (always close to hand, even if only ever dipped into), must have been among the first of them. (My only excuse for the oversight must be exhaustion from sleep deprivation.) The definition is in $\S7.5$ of Chapter I, and is clearly signposted in both the index and table of contents. It's on page 73 of the 1989 Springer reprint. Please repost your comment as an answer (with these details added, if you like), and I will probably accept it (certainly if no other answer comes along). $\endgroup$ – Calum Gilhooley Mar 13 at 18:19
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    $\begingroup$ No problem, I only remembered it because I had a topology final not so long ago, and Bourbaki was one of the references :) As you say, sometimes these 'common knowledge' concepts are so ingrained that we don't even think about having some reference at hand. $\endgroup$ – Guido A. Mar 13 at 18:55
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    $\begingroup$ @GuidoA. [+1 for the answer - I'll probably accept it in a couple of days.] The trouble is that, in my mind at least, a lot of vague pseudo-knowledge, and some outright nonsense, is also ingrained; so, when I can't find a reference for some idea, however intuitive, or some result, however obvious, I begin to worry, with good reason, whether it might not belong to the latter category, rather than the former! $\endgroup$ – Calum Gilhooley Mar 13 at 20:16
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Copying from the comments, Bourbaki's General Topology pt. I does this, although they first develop the language of filters and state it more as a remark than anything else. A definition can be found on page $73$ of the $1989$ Springer reprint, section $§7.5$.

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