A topological space is said to be a disconnected space if can be separated as the union of two non-empty disjoint open sets.
In other words, a topological space is said to be a disconnected space if there exist non-empty open sets and such that and . The pair is called the disconnection of .
Show that a two point discrete space is disconnected.
Let be a two point discrete space. If is any proper subset of , then both and are non-empty open subsets of such that and . This shows that is a disconnection of , so is a disconnected space.
• is disconnected.
• with upper limit topology is disconnected, since and are both open sets which form a disconnection of .
• Let be a non-empty set, with topology defined on . Then is a disconnected space.
• Every discrete space with more than one point is disconnected.