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?

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.

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 $§7.5$.