Components of a Space

A connected subspace of a topological space X is said to be the component of X if it is not properly contained in any connected subspace of X.

Note: The component of a connected space X is the whole space, X, itself.


The singleton subset of a two-point discrete space is its components.

Let X is be two-point discrete space, then the only possible connected subsets of X are its singleton subsets. As no singleton subset is properly contained in any other singleton subset, so these singleton subsets are components of X. Hence the singleton subsets of a two-point discrete space are its components.

• Let X be a topological space, then for each x \in X, there is exactly one component of X containing x.
• Let X be a topological space, then each connected subset of X is contained in a component of X.


Let X be a topological space, then every component is closed in X.


Let C be a component of X. If possible, suppose that C is not closed in X. Since \overline C is closed and connected and contains C, which is a contradiction to the fact that C is a component. Hence C = \overline C , so C is closed.