Connected Components and Totally Disconnected Spaces
Definition
Among the connected subspaces of a topological space $X$, a connected set that has only itself as a connected superset is called a connected component of $X$. In particular, the connected component containing $x \in X$ is written $C_{x}$. If every connected component of $X$ is a singleton, then $X$ is called a totally disconnected space.
Explanation
Connected Components
At first glance the definition seems to go in circles, but it is surprisingly not a big deal.
Since it would be far too many to call every connected space a ‘component’, if there is a connected set larger than a given set, we exclude the given one. You can think of it as continuously searching for such larger sets and taking whatever can be regarded as a ‘chunk’ as a single unit.
As a simple example, consider the Chinese character 休 (hyu); its left part 亻 and 木 are the connected components of 休. When talking about a certain connected component, there is no need to know or mention every connected component. If we regard whether there is a land route as connectivity, then for our country it suffices to say that the mainland, Jeju Island, Ulleungdo, Dokdo, and so on are the connected components.
Below are various properties of connected components. Most of them can be proved without much difficulty, but rather than focusing on that, it is important to make the effort to get familiar with them as facts.
Properties of Connected Components
- [1]: $x \in X$ belongs to exactly one $C_{x}$.
- [2]: For $a,b \in X$, either $C_{a} = C_{b}$ or $C_{a} \cap C_{b} = \emptyset$.
- [3]: Every connected space is a subset of some connected component.
- [4]: Every connected component of $X$ is a closed set in $X$.
- [5]: $X$ being a connected space is equivalent to $X$ having exactly one connected component.
Totally Disconnected Spaces
Meanwhile, a totally disconnected space may be thought of as the exact opposite concept of a connected space, where ’exact opposite’ does not mean negation. As you know, the negation of a connected space is merely a disconnected space, whereas a totally disconnected space is one that lacks connectedness with respect to ’every’ subset. A simple example is the discrete space.