I am reading Tao's Analysis II (third edition). Awesome book, if a bit lacking on graphical explanation.
On page 36, he gives the definition and an example:
Definition 2.4.1 (Connected spaces). Let (X, d) be a metric space. We say that X is disconnected iff there exist disjoint non-empty open sets V and W such that $V \cup W = X$. (Equivalently, X is disconnected if and only if X contains a non-empty proper subset which is simultaneously closed and open.) We say that X is connected iff it is non-empty and not disconnected.
Example 2.4.2. Consider the set $X := [1, 2] \cup [3, 4]$, with the usual metric. This set is disconnected because the sets [1, 2] and [3, 4] are open relative to X (why?).
I cannot seem to follow the reasoning for this. I know it has to do with the boundary points, balls, and relative topology of [1, 2] and [3, 4], but [1, 2] and [3, 4] "look" like closed sets to me (not open sets as is required by the definition).
Any help?
Best Answer
Openness and closedness depends on the ambient space. In the very common ambient space of the real numbers, $[1,2]$ is a non-open closed subset, as there does not exist any $r > 0$ such that we have open balls of radius $r$ such that $1 \in B_r(1) \subseteq [1,2]$. The same is true of $2 \in [1,2] \subseteq \mathbb{R}$ and $3,4 \in [3,4] \subseteq \mathbb{R}$.
However, for these sets as subspaces of the metric space $(X,d)$, we can see that setting $r = \frac{1}{2}$, we may construct such open balls $$1 \in B_{\frac{1}{2}}(1) \subseteq [1,2] \subseteq X \qquad \text{and} \qquad 2 \in B_{\frac{1}{2}}(2) \subseteq [1,2] \subseteq X\text{ .}$$ We may also do the same for $3,4 \in [3,4] \subseteq X$. This is because the elements of (for example) $B_{\frac{1}{2}}(1) \setminus [1,2] = (\frac{1}{2}, 1) \subseteq \mathbb{R}$ are simply not elements of the ambient metric space $(X,d)$.
Thus, in the context of the ambient metric space $X$, the sets $[1,2]$ and $[3,4]$ are open. They are still closed, as they are still equal to their closures.
A simple TikZ illustration follows: