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.