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 X is connected if it is not the union of two non-empty disjoint open sets.

Example:
            Every indiscrete space is connected.
            Let X be an indiscrete space, then X is the only non-empty open set, so we cannot find the disconnection of X. Hence X is connected.

Connected Subspace:
            A subspace Y of a topological space is said to connected subspace if Y is a connected as a topological space in its own right.

Example:
            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 A and B are open subsets of . Also  and . This shows that  is a disconnected subspace of .

Theorems:

• A topological space is connected if and only if 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 X and C is a connected subspace of X, Then C is contained either in A or in B.

Characterization of Connected Space:
            In a space, the following are equivalent:

• X is connected.
• The only open and closed subsets of X are , X.
• There does not exist a continuous map  from a space X onto the discrete space .