Intervals Are Connected

From Connectedness to Small Steps

The previous tutorial examined how a missing point between two members of a set can create a separation. In this tutorial, we look at intervals from another direction. The theorem that intervals are connected was established earlier in the course. Rather than re-prove it, we will explore a concrete feature of intervals that makes it easier to picture how their points fit together: any two points can be joined by finitely many sufficiently small steps that stay inside the interval.

This small-step property is useful in its own right, but it is not another definition of connectedness. In fact, a set can have the small-step property and still be disconnected. Keeping those ideas separate helps clarify what the connectedness theorem says, and what it does not say. We will also use interval connectedness to prove that a locally constant function on an interval cannot take different values at different points.

The word “finite” matters: no matter how small the allowed step size is, the chain must reach its destination after finitely many steps. If \(x=y\), the one-point list \(x_0=x\) is a chain with no steps. If \(E\) is empty, the condition that every pair of its points can be joined holds vacuously.

Every Interval Allows Small-Step Travel

For an interval, the construction is direct. Given two points \(x\) and \(y\), divide the segment between them into equal pieces. The points produced by the division lie between \(x\) and \(y\), so the defining property of an interval keeps every one of them inside the interval. Making enough pieces ensures that each step is smaller than the chosen tolerance.

Proof. If \(x=y\), the one-point list \(x_0=x\) is an \(\varepsilon\)-chain. Now suppose \(x\ne y\). Choose a positive integer \(n\) such that \(n>|y-x|/\varepsilon\), and define \(x_k=x+\frac{k}{n}(y-x)\) for \(k=0,1,\ldots,n\). Each \(x_k\) lies between \(x\) and \(y\), since \(0\leq k/n\leq1\). Because \(J\) is an interval and \(x,y\in J\), it follows that \(x_k\in J\) for every \(k\). Also, for \(k=1,\ldots,n\), \(|x_k-x_{k-1}|=|\frac{1}{n}(y-x)|=|y-x|/n<\varepsilon\), where the last inequality follows from the choice of \(n\). Thus \(x_0,\ldots,x_n\) is the required \(\varepsilon\)-chain in \(J\). \(\square\)

The proof used only the interval property and the ability to choose an integer \(n\) larger than a specified real number. It did not depend on the interval being bounded, closed, or open. For example, the same subdivision argument applies to \((-\infty,7)\), \([3,\infty)\), and \(\mathbb{R}\). The endpoints of an interval matter for membership at the endpoints, but they do not change the small-step construction between two points already in the interval.

Small-Step Chains Are Not the Same as Connectedness

The chain theorem gives a useful picture of how one can travel through an interval, but it should not be mistaken for a characterization of connectedness. The rational numbers provide a cautionary example. Between two rational numbers, one can choose equally spaced rational points, with each step as small as desired. Thus \(\mathbb{Q}\) is \(\varepsilon\)-chain connected for every \(\varepsilon>0\). Nevertheless, \(\mathbb{Q}\) is not connected as a subset of \(\mathbb{R}\): a cut at an irrational number separates its rational points into nonempty left and right pieces.

This comparison highlights a useful distinction. The interval property says that every real number between two points of the set also belongs to the set. The chain property asks only for a finite route through points of the set, with consecutive points close together. Those are different requirements. Intervals have both properties, but the chain property alone does not guarantee connectedness.

Locally Constant Functions on Intervals

A function is locally constant if, around each point of its domain, there is a relative neighborhood on which the function takes just one value. On an interval, it cannot switch from one value to another. This consequence follows from connectedness, not from the chain construction alone.

Proof. If \(J=\varnothing\), the conclusion that \(f\) is constant is vacuous. Suppose \(J\ne\varnothing\), and choose \(a\in J\). Set \(A=\{x\in J:f(x)=f(a)\}\). Then \(a\in A\), so \(A\ne\varnothing\). We show that both \(A\) and \(J\setminus A\) are relatively open in \(J\). If \(x\in A\), local constancy gives an open set \(U\) containing \(x\) such that \(f(z)=f(x)=f(a)\) for every \(z\in J\cap U\). Hence \(J\cap U\subseteq A\), so \(A\) is relatively open in \(J\). If \(x\in J\setminus A\), then \(f(x)\ne f(a)\). Local constancy gives an open set \(V\) containing \(x\) such that \(f(z)=f(x)\) for every \(z\in J\cap V\). No such \(z\) can belong to \(A\), because \(f(z)=f(x)\ne f(a)\). Thus \(J\cap V\subseteq J\setminus A\), so \(J\setminus A\) is relatively open as well. If \(J\setminus A\) were nonempty, \(A\) and \(J\setminus A\) would be disjoint, nonempty, relatively open sets whose union is \(J\), contradicting the connectedness of intervals established earlier. Therefore \(J\setminus A=\varnothing\), so every \(x\in J\) satisfies \(f(x)=f(a)\). Hence \(f\) is constant on \(J\). \(\square\)

Why These Two Viewpoints Matter

The small-step theorem gives a constructive way to move through an interval: subdivide the segment joining the chosen points, and use the interval property to keep the subdivision points in the set. The locally constant function theorem gives a different consequence of interval connectedness: no locally constant assignment can change value across the interval. The first statement is about the geometry of points and distances; the second is about how values can vary across a domain.

A common pitfall is to infer connectedness from the existence of chains at every scale. The rational example shows why that inference fails: chains can jump across a cut in the real line without containing the cut point itself. When the goal is to prove connectedness, use a valid connectedness criterion, such as ruling out a separation. When the goal is to find a finite path with controlled step sizes, the chain construction is the appropriate tool.