# Definition of limit involving open sets

I recently had a bit of an obsession with the definition of limits, and that made think about this function: $$f: \mathbb{Q} \to \mathbb{R}; \ \ \ \ f(x) = x$$

What is $$\ \lim_{x \to 0} f(x)\$$? Intuitively, I think $$\ \lim_{x \to 0} f(x) = 0\$$. I then looked at the definition provided by Wikipedia:

According to this definition, $$\ \lim_{x \to 0} f(x)\$$ does not exist; $$\mathbb{Q}$$ does not contain within it any open intervals whatsoever, let alone ones required by the definition. However, there is also a definition for metric spaces:

According to this definition, the limit is indeed $$\ \lim_{x \to 0} f(x) = 0\$$; setting $$X, Y, N = \mathbb{R}$$ and $$M=\mathbb{Q}$$ makes showing that result trivial. This got me wondering, why does there seem to be a requirement for an open interval (or open ball) in the first definition, and not the second?

The idea that I heard was that, the first definition has that restriction to select for only well behaving functions. The second definition captures all functions that converge to a limit, even ones that don't behave well. That seems to make sense.

I then scroll down to the topological definition, I see the following:

This too seems to agree with the metric space definition, because with $$\Omega = \mathbb{Q}\$$, $$\ U \cap \Omega$$ eliminates any non-rational numbers by definition. However, I also heard that (for the topological definition) there should exist such an open set $$Op \subset \mathbb{R}$$, so that $$0 \in Op$$ and $$\mathbb{Q}$$ contains $$Op \setminus \{ 0 \}$$.

I don't know if this requirement is accurate (I heard it on reddit). Is this requirement accurate?

If it is accurate, I have a follow-up question. With the standard topology on $$\mathbb{R}$$, the subspace $$\mathbb{Q}$$ itself is not an open set, and neither can it contain within it any open sets. So what gives? How is it that the requirement for $$\mathbb{Q}$$ containing an open set is still met, somehow?

• You should not necessarily trust what you read on Wikipedia since anybody can edit no matter how little do they know about the subject, textbooks are better as a source. The general topological and metric definitions of limits are fine. The fact that $\mathbb Q$ is not open in $\mathbb R$ is irrelevant. Mar 1 at 16:02
• In fact, if you continue reading Wikipedia's article on limits of functions (section "More general definition using limit points and subsets") you will get to the definition consistent with the topological one. Mar 2 at 1:49