I'm trying to understand a point made in a set of lecture notes on topological spaces. It takes $X = \{p,q\}$, and declares that the open sets are $\emptyset$, $\{p\}$, and $\{p,q\}$. These certainly satisfy the axioms for topology. The comment in the notes are:
Roughly speaking, this means that "one point is infinitely close to the other, but not vice versa," which is not something that would normally a rise in a geometric setting.
I'm struggling to understand what this means. This is surely talking about $p$ and $q$. I think it is trying to say that the fact that $\{p,q\}$ is open implies one of $p$ and $q$ is "infinitely close to the other," but the fact that $\{p\}$ itself is open implies it isn't "infinitely close to $q$." I'm struggling to understand this characterization of openness.
Best Answer
The fact that $\{p,q\}$ is open doesn't tell you anything--the whole space is always open. Rather, it is the fact that $\{q\}$ is not open which tells you $p$ is "infinitely close to $q$". Indeed, this says that every neighborhood of $q$ contains $p$ (since the only such neighborhood is $\{p,q\}$). If you think of neighborhoods as containing all "sufficiently close points", this means that $p$ is always "sufficiently close" to $q$, no matter how close you want it to be.
On the other hand, since $\{p\}$ is open, $q$ is not "infinitely close to $p$". There is some neighborhood of $p$ that does not contain $q$, so $q$ is "far enough from $p$" to be outside that neighborhood.