Connectedness and components

Try

Removing one point from the real line separates it into two open half-lines. Removing one point from a circle leaves a connected arc-like space. This difference will later rule out a homeomorphism between the circle and an interval.

Connectedness means a space cannot be split into two disjoint nonempty relatively open pieces covering it. It is not a claim that every pair of points can be joined by a path; that stronger property is treated next.

Predict

Predict whether a connected set remains connected after taking its closure. It does. Any separation of the closure would have one side missing the original connected set, but a nonempty relative open in the closure must meet that dense set.

Predict whether connected components must be open. They are always closed, but openness requires additional local connectedness assumptions. In Q with its usual topology, components are singletons and none is open.

Experiment

Compare standard component structures.

Space	Connected components
R	One component
R without zero	Two half-lines
Discrete three-point space	Three singletons
Q with usual subspace topology	Singletons

For Q, separate any two distinct rationals using an irrational cut between them. This proves a component cannot contain both.

Understand

A separation of X is a pair of disjoint nonempty opens whose union is X. Equivalently, X has a nonempty proper clopen subset: either side of a separation is also closed because its complement is open. Connectedness means no such separation exists.

A continuous image of a connected space is connected. A separation of the image would pull back to a separation of the source, with nonemptiness supplied by surjectivity onto the image.

A union of connected subsets sharing a common point is connected. In a proposed separation, the common point belongs to one side. Each connected subset must lie entirely on that side, because otherwise its intersections with the two sides would separate it. The other side of the union is then empty.

Define the component of x as the union of all connected subsets containing x. The common-point theorem proves it connected and maximal. Two components that intersect have connected union, so maximality makes them equal. Hence components partition the space.

The closure of a connected set is connected: the original set lies on one side of any proposed separation of the closure, but the other nonempty relatively open side must meet the dense original set. Therefore a component’s closure is connected and maximality forces the component to equal its closure. This proves every component is closed without claiming it is open.

Apply

Prove each connected subset of Q has at most one point. Given two rationals p<q, choose an irrational r between them. The rational points below and above r form disjoint relative opens covering Q, and they separate any subset containing both p and q.

Repair “The components form an open cover.” They form a partition by closed connected sets in every space. They become open under local connectedness, a hypothesis that cannot be inferred from a familiar drawing or from the word component.

Continue

Path connectedness provides a constructive way to prove connectedness but is not equivalent in general. Optional depth: prove that every connected subset of R is an interval, using a missing intermediate point to create a separation. Combine that with compactness to classify nonempty compact connected real subsets as closed intervals, including singleton intervals.