Definition 3

The component (or connected component) of a point of a topological space $X$ is the largest connected subset of $X$ which contains this point. The components of a subset $A$ of $X$ are the components of the points of $A$, relative to the subspace $A$ of $X$.

If a space is connected, the component of each point is the whole space. If a space $X$ is such that for each pair $(x, y)$ of points of $X$ there is a connected set containing $x$ and $y$, then $X$ is connected.

A space $X$ is said to be totally disconnected if the component of each point of $X$ consists of the point alone. A subset $A$ of $X$ is a totally disconnected set if the subspace $A$ of $X$ is totally disconnected.

A discrete space is totally disconnected, but one should beware of confusing these two notions; for example, we shall see in Chapter IV, § 2, no. 5, that the rational line, which is not a discrete space, is totally disconnected.

A set which is both open and closed contains the component of each of its points, whence the component of a point is contained in the intersection of the sets which are both open and closed and which contain this point. However, the component of a point is not necessarily equal to this intersection (cf. Exercise 9 and Chapter II, § 4, no. 4, Proposition 6).