The Structure of Connected Subsets of R

From Connected Sets to Their Components

The Characterization of Connected Subsets of \(\mathbb{R}\) gives a direct test: a subset of the real line is connected exactly when it is an interval. This turns the study of connectedness into a question about order. Whenever a point lies between two points of an interval, it must also belong to that interval.

For a set that is not itself an interval, its connected components describe the largest connected pieces it contains. The Components Partition a Set theorem says that these pieces are pairwise disjoint and cover the set. The Maximality Characterization of Components says that each is a maximal connected subset. We will use these earlier results to examine additional structure: what happens at the endpoints of a component, and what this implies when the original set is closed.

Since a connected subset of \(\mathbb{R}\) is an interval, every component is an interval. But components need not all have the same endpoint form. A component may include one endpoint and omit the other, include both, or omit both. The membership of its endpoints is controlled by the set \(E\) itself.

Finite Endpoints of Components

Components are relatively closed in their containing set, by the Components Are Relatively Closed theorem. This does not say that a component must be closed in all of \(\mathbb{R}\): the containing set might not be closed. It does, however, determine exactly what happens at a finite endpoint of a component.

Proof. First consider the lower endpoint \(\alpha=\inf C\). If \(\alpha\in C\), then \(C\subseteq E\) gives \(\alpha\in E\).

Conversely, suppose \(\alpha\in E\). For each positive integer \(n\), the definition of infimum gives a point \(c_n\in C\) such that \(\alpha\leq c_n<\alpha+1/n\). Hence \(c_n\to\alpha\). Because \(C\) is closed relative to \(E\), its relative closure satisfies \(\overline{C}\cap E=C\). The convergence \(c_n\to\alpha\) implies \(\alpha\in\overline{C}\), and the assumption \(\alpha\in E\) then gives \(\alpha\in C\).

For the upper endpoint \(\beta=\sup C\), if \(\beta\in C\), then \(\beta\in E\). If instead we assume \(\beta\in E\), the definition of supremum gives, for each positive integer \(n\), a point \(d_n\in C\) with \(\beta-1/n<d_n\leq\beta\). Thus \(d_n\to\beta\). Relative closedness gives \(\beta\in\overline{C}\cap E=C\). This proves both endpoint claims. \(\square\)

The theorem is useful because it rules out endpoint choices that might otherwise seem possible. If a finite endpoint of a component belongs to \(E\), it must belong to the component. If it is not in the component, then it is not in \(E\). In particular, a component cannot stop just short of a point of \(E\) at its endpoint.

Components of Closed Sets

For an arbitrary set \(E\), its components are relatively closed in \(E\), but may fail to be closed in \(\mathbb{R}\). When \(E\) is closed in \(\mathbb{R}\), relative closedness becomes ordinary closedness. Combining that fact with the interval characterization gives a particularly simple description.

Proof. Let \(C\) be a connected component of the closed set \(E\). By the Components Are Relatively Closed theorem, \(C\) is closed relative to \(E\). To show it is closed in \(\mathbb{R}\), take any \(x\in\overline{C}\), where the closure is in \(\mathbb{R}\). Since \(C\subseteq E\) and \(E\) is closed, \(\overline{C}\subseteq E\), so \(x\in E\). Relative closedness now gives \(x\in C\). Thus \(\overline{C}\subseteq C\); the reverse inclusion always holds, so \(C=\overline{C}\) and \(C\) is closed in \(\mathbb{R}\).

A component is connected by definition. The Characterization of Connected Subsets of \(\mathbb{R}\) therefore says that \(C\) is an interval. Since \(C\) is also closed in \(\mathbb{R}\), it is a closed interval, allowing an unbounded interval or a singleton. \(\square\)

“Closed interval” here includes intervals such as \([a,b]\), \([a,\infty)\), \((-\infty,b]\), and \(\mathbb{R}\), as well as \([a,a]=\{a\}\). It does not mean that every component of a closed set is bounded. Nor does it say that the whole closed set is one interval: a closed set may have many components.

Missing Points and the Shape of Components

A missing point between two members of a set is an obstruction to putting those members in the same connected subset. Indeed, any connected subset of \(\mathbb{R}\) containing \(x<y\) must contain the entire interval \([x,y]\). If \(c\) lies strictly between \(x\) and \(y\) and \(c\notin E\), no connected subset of \(E\) can contain both \(x\) and \(y\).

This is a convenient way to identify components: first find intervals contained in the set, then check whether missing points prevent those intervals from being enlarged. It also explains why a single omitted point can split a set even when the parts on either side approach that point as closely as desired. A positive-width gap is not required.

Reading the Structure Carefully

The component description depends on the set and on the order of the real line. A component is an interval because connected subsets of \(\mathbb{R}\) are intervals. It is maximal among intervals contained in the set because it is a maximal connected subset. Its finite endpoints are included precisely when they belong to the set. If the set is closed, relative closedness upgrades each component to a closed interval.

These facts should not be confused with the claim that distinct components must be separated by a gap of positive length. In the example \(\mathbb{R}\setminus\mathbb{Z}\), neighboring components are separated by an omitted integer, but their distance is zero: points in the two intervals can be chosen arbitrarily close to that integer. The relevant obstruction is the missing point and the interval property, not positive distance.

For an open subset of \(\mathbb{R}\), the earlier Components of Open Sets Are Open Intervals theorem gives the corresponding open-set description: its components are open intervals, possibly unbounded. For a closed subset, the theorem proved here gives closed intervals. For a set that is neither open nor closed, endpoint inclusion can vary from component to component, as the endpoint theorem and the first example show.