📂Topology

Connected Components and Totally Disconnected Spaces

Definition

A connected component of a topological space $X$ is a connected subset that does not have a connected superset other than itself among the connected subsets of $X$. In particular, the connected component that includes $x \in X$ is denoted as $C_{x}$. If all connected components of $X$ are singleton sets, then $X$ is called a Totally Disconnected Space.

Description

Connected Components

At first glance, the definition seems to go in circles, but it’s actually quite a straightforward concept.

Since there are too many potential “components” to consider every connected space a component, any connected set that’s larger than itself is excluded. Think of it as finding the largest “clumps” that can be considered as a single unit.

For a simple example, consider the Chinese character for “rest”, 休. The left parts, 亻(person) and 木(tree), are the connected components of rest. When talking about connected components, there is no need to know or mention all the connected components. If connectivity is considered in terms of land connections, then South Korea’s main land, Jeju Island, Ulleungdo, and Dokdo can sufficiently be called connected components.

Below are several properties of connected components. Most can be proven without difficulty but familiarizing oneself with these as facts is more important than focusing on the proofs.

Properties of Connected Components

• [1]: $x \in X$ belongs to only one $C_{x}$.
• [2]: Regarding $a,b \in X$, it is either $C_{a} = C_{b}$ or $C_{a} \cap C_{b} = \emptyset$.
• [3]: Every connected space is a subset of some connected component.
• [4]: All connected components of $X$ are closed sets in $X$.
• [5]: $X$ being a connected space is equivalent to $X$ having only one connected component.

Totally Disconnected Space

On the other hand, the concept of a totally disconnected space can be considered the opposite of a connected space, though “opposite” here does not imply negation. As known, the negation of a connected space is merely a disconnected space, while a totally disconnected space lacks connectivity in ‘all’ its subsets. A simple example would be a discrete space.