Why is the topological definition of function limit absent from textbooks?

Question

Am reading the topological definition of limits on Wiki:

Suppose $X,Y$ are topological spaces with $Y$ a Hausdorff space. Let $p$ be a limit point of $Ω \subset X$, and $L \in Y$. For a function $f : Ω \to Y$, it is said that the limit of $f$ as $x$ approaches $p$ is $L$ (i.e., $f(x) \to L$ as $x \to p$) and written

$ \lim_{x \to p}f(x) = L $

if the following property holds: For every open neighborhood $V$ of $L$, there exists an open neighborhood $U$ of $p$ such that $f(U \cap Ω - \{p\}) \subset V$.

I really like the generality of this definition, but somehow and can't find any english textbooks that mentions this definition (Munkres, Kelley, Willard, etc.), except Bourbaki.

Do you know any modern book/reference where this definition is treated in details?

That seems like a pretty common definition, though most introductory texts probably stick to the category of metric spaces. — anomaly, Jun 05 '20 at 11:43
As well as the duplicate question cited, see Does any book now in print define the meaning of $\lim_{x\to a}f(x)=b$ for $f\colon E\to Y$, $E\subseteq X$, $X$ a topological space, $Y$ Hausdorff?. — Calum Gilhooley, Jun 05 '20 at 12:16
@CalumGilhooley I see you asked a similar question! So apart from Bourbaki did you find any other source? — Alphie, Jun 05 '20 at 12:24
Apart from the out-of-print book by Schubert (which in any case seems to be closely based on Bourbaki in respect of this definition), I think I still haven't come across the definition anywhere else. — Calum Gilhooley, Jun 05 '20 at 12:28

score 2 · Answer 1 · answered Jun 05 '20 at 11:43

2

Not a book, but this General Topology paper is giving a good introduction on the topic pages 16 onward.

answered Jun 05 '20 at 11:43

mathcounterexamples.net

70,018

(+1) In case the URL expires, it would be a good idea to give the title of the lecture notes and the name of the author, and quote the definition (which in any case is quite short). – Calum Gilhooley Jun 05 '20 at 12:49

Why is the topological definition of function limit absent from textbooks?

1 Answers1