A topological space which cannot be written as the union of two non-empty disjoint open sets is said to be a connected space.
In other words, a space is connected if it is not the union of two non-empty disjoint open sets.
Every indiscrete space is connected.
Let be an indiscrete space, then is the only non-empty open set, so we cannot find the disconnection of . Hence is connected.
A subspace of a topological space is said to be a connected subspace if is connected as a topological space in its own right.
The subset of a connected space may not be connected.
The set of real numbers cannot be represented as the union of two disjoint non-empty sets, so is a connected space.
Next suppose that and
Since and are open subsets of , so and are open subsets of . Also and . This shows that is a disconnected subspace of .
• A topological space is connected if and only if it cannot be represented as the union of two disjoint non-empty closed sets.
• An infinite set with co-finite topology is a connected space.
• Any continuous image of a connected space is connected.
• The range of a continuous real unction defined on a connected space is an interval.
• If is a disconnection of and is a connected subspace of , then is contained either in or in .
Characterization of a Connected Space
In a space, the following are equivalent:
• is connected.
• The only open and closed subsets of are , .
• There does not exist a continuous map from a space onto the discrete space .