Intuition/explanation of totally disconnected sets (or subsets of R).

Question

I'm trying to understand Rudin's solution to an analysis problem (Construction of a Borel set with positive but not full measure in each interval) and the term totally disconnected is not something I've seen. Resources have been unhelpful. Could someone explain the intuition behind this property (in regards to $\mathbb{R}$ or Cantor set, for example)? Thank you in advance.

Answer 1

Totally disconnected is sort of opposite to connected.

Being connected is sometimes a counterintuitive concept. So let's have a look at a different, yet similar and more intuitive idea: path connectedness.

A space is path connected if any two points can be connected via continuous path. Examples are: $\mathbb{R}$, the sphere $S^n$ and a singleton $\{*\}$. It is easy to write the opposite of that: a space it totally path disconnected if no two distinct points can be connected via a path. Examples are: $\mathbb{Q}$, the Cantor set, any discrete space (note funny fact: single point is the only nonempty space that is both connected and totally disconnected at the same time). An example of something in-between is $[0,1]\cup[2,3]$.

I often imagine it like this: if I land on a totally disconnected space then regardless of the landing point I cannot move anywhere. I have to jump.

We can use this idea for normal connectedness as well. Except that first we have to somehow translate "connectedness" into a relationship between points. And we do that by realizing that a space is connected if and only if any two points belong to a connected subset. This gives us a way to introduce the opposite: a space is totally disconnected if no two distinct points lie in a connected subset.

Examples I gave you earlier are examples for connected and totally disconnected spaces as well (although these concepts do not pairwise coincide).

Read more here: https://en.wikipedia.org/wiki/Totally_disconnected_space

Answer 2

It helps to recall the definition of connected first. A space $X$ is connected if it's impossible to find disjoint, non-empty opens $U$ and $V$ so that $X = U\cup V$. So $X$ is disconnected if there are some points $x,y\in X$ and disjoint open neighbourhoods $U\ni x$ and $V\ni y$ so that $X = U\cup V$.

Totally disconnected spaces take this further. $X$ is totally disconnected if for every $x$ and $y$ there are $U$ and $V$ as above.

Answer 3

As the term says, totally disconnected literally means that every point is not "connected" to any other point (possibly, the notion of totally path-disconnected is easier to visualize: no two points can be joined by a continuous path). A good example to keep in mind is $\mathbb{Q}$ (with the topology inherited from the real line). For every two rational numbers $a$ and $b$, you can find an irrational number $\tau \in (a,b)$, so that

$$[a,b] \cap \mathbb{Q} = ([a,\tau] \cap \mathbb{Q}) \sqcup ([\tau,b] \cap \mathbb{Q}).$$

This shows that every non-empty, closed interval in $\mathbb{Q}$ can be decomposed into two disjoint, non-empty, closed intervals, which is not possible, for instance, in the real line. And you can repeat this procedure again and again: intervals disintegrate in your hands, until (in the limit) you are left with just points.

Answer 4

Totally disconnected $X$ means that the canonical map $X \to \pi^0 (X)$ is an homeomorphism.