Disconnected Space

A topological space $X$ is said to be a disconnected space if $X$ can be separated as the union of two non-empty disjoint open sets.

In other words, a topological space $X$ is said to be a disconnected space if there exist non-empty open sets $A$ and $B$ such that $A \cap B = \phi$ and $A \cup B = X$. The pair $\left\{ {A,B} \right\}$ is called the disconnection of $X$.

Example:

Show that a two point discrete space is disconnected.

Let $X$ be a two point discrete space. If $A$ is any proper subset of $X$, then both $A$ and ${A^c}$ are non-empty open subsets of $X$ such that $A \cap {A^c} = \phi$ and $A \cup {A^c} = X$. This shows that $\left\{ {A,{A^c}} \right\}$ is a disconnection of $X$, so $X$ is a disconnected space.

Examples:
• $\left( {0,1} \right) – \left\{ {\frac{1}{2}} \right\}$ is disconnected.
• $\mathbb{R}$ with upper limit topology is disconnected, since $\left\{ {x:x > a} \right\}$ and $\left\{ {x:x \leqslant a} \right\}$ are both open sets which form a disconnection of $\mathbb{R}$.
• Let $X = \left\{ {a,b,c} \right\}$ be a non-empty set, with topology $\tau = \left\{ {\phi ,X,\left\{ c \right\},\left\{ {a,b} \right\}} \right\}$ defined on $X$. Then $\left( {X,\tau } \right)$ is a disconnected space.
• Every discrete space with more than one point is disconnected.