Disconnected Space

A topological space X is said to be 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.


Show that two point discrete spaces 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.

\left( {0,1} \right) - \left\{ {\frac{1}{2}} \right\} is a 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.