A space that is not connected still falls into pieces that are, and the pieces can be named: the component of a point is the largest connected subset containing it. Every point lies in exactly one component, so the components cut the space up without overlap, and how many there are is a topological invariant — often the quickest way to show two spaces are not homeomorphic.

An archipelago has islands. Each is as large a piece of land as you can walk without leaving it, no island overlaps another, and counting them tells you something about the archipelago that no map projection can change.

A space in three components. Each is connected, each is as large as a connected subset containing its points can be, and no two overlap — because two connected sets sharing a point would unite into a larger connected set, contradicting the largeness of both. The count is an invariant: a homeomorphism carries components to components.

The component of $x$ is the union of every connected subset containing $x$. That union is connected, by the theorem on sets sharing a point, and nothing connected containing $x$ is larger — so it is the largest one. Two components either coincide or share no point, and every point lies in its own, so the components partition the space.

• Components are closed: the closure of a connected set is connected and contains the component, which is largest, so the component equals its own closure.
• Components need not be open. In $\mathbb{Q}$ every component is a single point, and a single point is not open there — a space whose components are all points is called totally disconnected.
• A connected space has exactly one component, the whole space. A discrete space has one for each point, and there they are open as well as closed.
• The number of components is a topological invariant. $\mathbb{R}$ has one and $\mathbb{R}\setminus\{0\}$ has two, which is the cleanest proof that removing a point from the line changes it as a space.
• Path components are defined the same way with paths in place of connected subsets. They sit inside the components and can be strictly smaller, in exactly the spaces where connected and path connected part company.

Take two components with a point in common. Both are connected, and connected sets sharing a point unite into a connected set, so their union is connected. That union contains the first component and contains the point the first is the component of — and the component of a point is the largest connected set containing it, so the union cannot be larger than the first component. It therefore equals it, which forces the second component inside the first. The same argument with the two swapped puts the first inside the second, so they are equal. Since every point lies in its own component, the components cover the space, and this shows any two of them are equal or share nothing.

Suppose two components share a point.

Both are connected and they share a point, so their union is connected.

The union is a connected set containing the first point, and the component of that point is the largest such set.

The same argument at the second point puts the union inside the second component.

Each is contained in the other, so the two components coincide.

The component of a point is the union of all connected subsets containing it, and that union is itself connected.

• The component of a point is the largest connected subset containing it.
• Components are closed, cover the space, and no two of them overlap.
• They need not be open: in $\mathbb{Q}$ every component is a point.
• Their number is a topological invariant.

Great work! You now know how to:

• find the component of a point as the largest connected set through it;
• see that components are closed, disjoint and cover the space;
• count components to tell two spaces apart;
• keep path components and components apart.