Connected and Disconnected Sets In Topological Spaces

Connected and Disconnected Sets In Topological Spaces

Recall from the Connected and Disconnected Topological Spaces page that a topological space $X$ is said to be disconnected if there exists open sets $A, B \subset X$ such that $A, B \neq \emptyset$, $A \cap B = \emptyset$ and:

(1)
\begin{align} \quad X = A \cup B \end{align}

In such cases, we cal $\{ A, B \}$ a separation of $X$. Furthermore, the topological space $X$ is said to be connected if it is not disconnected.

We now turn our attention to defining connected and disconnected sets in a topological space.

 Definition: Let $X$ be a topological space and let $A \subseteq X$. $A$ is said to be Connected if it is a connected topological space with respect to the subspace topology on $A$. Similarly, $A$ is said to be Disconnected if it is a disconnected topological space with respect to the subspace topology on $A$.

For example, consider the topological space $\mathbb{R}$ with the usual topology. Let $A = (0, 1) \cup [2, 3]$. Then the set $A$ (with the subspace topology) is a disconnected topological space. Let $B = (0, 1)$ and $C = [2, 3]$. Clearly $B, C \subset A$, $B, C \neq \emptyset$, $B \cap C = \emptyset$, and $A = B \cup C$. So, we only need to show that $B$ and $C$ are open in $A$.

Since $A$ has the subspace topology, consider the open sets $(0, 1)$ and $\left ( \frac{3}{2}, 4 \right )$ in $\mathbb{R}$. When we intersect these open sets with $A$ we see that:

(2)
\begin{align} \quad A \cap (0, 1) = (0, 1) = B \end{align}
(3)
\begin{align} \quad A \cap \left ( \frac{3}{2}, 4 \right ) = [2, 3] = C \end{align}

Therefore $B$ and $C$ are open in $A$. This shows that $\{ B, C \}$ is a separation of $A$ and that $A$ is a disconnected set.