Connectedness

Compactness (2B.3 Compactness) says a space is "small" in a precise sense. Connectedness says it is "in one piece". The idea is simple enough to state in a sentence, and it is the source of the intermediate value theorem, which in 2A.9 Continuous Functions seemed to depend on special properties of the real line. Here it is reborn as a statement about any connected space: a continuous real-valued function on a connected space can't skip values.

Connectedness is also the first topological invariant in the guidebook: a property that survives any continuous deformation with a continuous inverse. Counting the pieces left over after removing a point is already enough to prove that a line and a plane are different spaces. That kind of argument, finding a quantity that can't change under deformation, is how topology tells spaces apart, and it leads to the fundamental group (7A.4 The Fundamental Group) and to the Poincaré conjecture itself (7A.9 The Poincaré Conjecture, Precisely). It is a shorter chapter than the last three, because the ideas are simpler, but they are used constantly.

By the end of this chapter you will be able to:

• prove that a set is connected or disconnected, and that the connected subsets of $\mathbb{R}$ are exactly the intervals;
• use the general intermediate value theorem, and the "open and closed" argument that underlies it;
• prove that path-connected sets are connected, and explain why the converse fails;
• find the connected components of a space, and explain how a power grid splits into islands;
• use connectedness to show that two spaces are not homeomorphic.

4.1A grid that falls apart

The point of the example is that "being in one piece" is not a vague idea. It is computable, it can change abruptly when connections are cut, and it governs what happens next. The rest of the chapter makes it precise for metric spaces, where the pieces need not be finite.

4.2Connected spaces

A space should count as disconnected if it splits into two parts, neither of which comes arbitrarily close to the other. "Doesn't come close" is said precisely by openness.

If $X = U \cup V$ as in the definition, then $U = X \setminus V$ is also closed. So $X$ is connected exactly when its only subsets that are both open and closed (clopen) are $\varnothing$ and $X$. This reformulation is the one used in proofs.

For a subset $Y \subseteq X$, openness is relative to $Y$ (2B.1 Metric Spaces). The set $Y = [0, 1] \cup [2, 3]$ is disconnected: $[0, 1]$ and $[2, 3]$ are both open in $Y$, though not in $\mathbb{R}$.

Look at where completeness entered: the supremum $s$. Over $\mathbb{Q}$ the same proof fails, and indeed $\mathbb{Q}$ is disconnected: $\{q < \sqrt2\}$ and $\{q > \sqrt2\}$ split it into two open pieces.

4.2.1The intermediate value theorem, reborn

On $X = [a, b]$ this is the IVT of 2A.9 Continuous Functions. The new version works on any connected space. A continuous temperature on the surface of the Earth, which is connected, takes every value between the coldest and the hottest. And a continuous function on a connected space with values in $\mathbb{Z}$ must be constant, because its image is an interval of integers containing no non-integers between its points. This last remark is a tool used everywhere: to prove that an integer-valued quantity (a degree, a winding number, a count of solutions) can't change under continuous deformation, show it is continuous (7A.7 Smooth Topology).

4.2.2The open-and-closed argument

The proof of Theorem 4.2 has a pattern that generalises into a method of proof.

The method turns a global statement into three local ones: one example, stability under small perturbations (openness), and stability under limits (closedness). It is how one proves that a differential equation has a solution for all times in an interval (2B.10 Ordinary Differential Equations), how estimates are propagated along a Ricci flow ("the set of times at which the bound holds is open and closed", 11A.4 Maximum Principles under Ricci Flow), and, as the continuity method of PDE, how equations are solved by deforming an easy one into a hard one (6A.7 Nonlinear Parabolic Equations). Exercise 4.18 is a rehearsal.

4.3Path-connected spaces

A more intuitive notion is that any two points can be joined by a continuous path.

Path-connected spaces are everywhere. A convex subset of $\mathbb{R}^n$, one that contains the segment between any two of its points, is path-connected by straight lines: balls of every metric of 2B.1 Metric Spaces are convex. Spheres $S^n$ ($n \geq 1$) are path-connected along great circles. $\mathbb{R}^n$ with a point removed is path-connected if $n \geq 2$: go around the missing point.

The converse of Proposition 4.6 is false, and the standard counterexample is worth seeing once.

For open sets of $\mathbb{R}^n$, which is the case that matters most in analysis, the two notions coincide, and the proof is another use of the open-and-closed argument.

A connected open subset of $\mathbb{R}^n$ is called a domain, and PDE is mostly done on domains or on connected manifolds. The reason is the open-and-closed argument again: for instance, a function with zero gradient on a domain is constant (2B.8 Calculus in Several Variables), but on a disconnected open set it need only be constant on each piece.

4.4Connected components

If a space is not connected, it is natural to break it into maximal connected pieces.

Two components are either equal or disjoint (if they shared a point, their union would be connected and larger). So the components partition $X$, and "lies in the same component as" is an equivalence relation in the sense of 2A.2 Sets, Functions and Equivalence. Components are closed, because the closure of a connected set is connected (Exercise 4.13). They need not be open: each point of $\mathbb{Q}$ is its own component.

For a finite network, the same definitions reduce to the familiar ones for graphs: draw each node as a point and each edge as a segment, and the connected components of the resulting space are exactly the groups of nodes that can reach each other along edges. Finding them is a basic algorithm: start at a node, visit every neighbour, every neighbour's neighbour, and so on, until nothing new is found. That is the open-and-closed argument run by a computer.

4.5Connectedness as an invariant

Homeomorphic spaces are the same "up to continuous deformation". The open interval $(0, 1)$ is homeomorphic to $\mathbb{R}$ (by $x \mapsto \tan(\pi(x - \tfrac12))$), a square to a disc, and the surface of a cube to a sphere. A homeomorphism carries open sets to open sets, so it carries connected sets to connected sets, compact to compact, and components to components. Properties preserved by every homeomorphism are called topological.

Completeness is not topological: $\mathbb{R}$ is complete, $(0, 1)$ is not, and they are homeomorphic. Completeness depends on the metric, not just on which sets are open. Compactness and connectedness, by contrast, can both be defined using only open sets (by covers, 2B.3 Compactness, and by clopen sets, above), and so they are topological.

The same trick shows that a circle is not homeomorphic to an interval (removing an interior point disconnects the interval but not the circle), and that the letter X is not homeomorphic to the letter T (Exercise 4.15). But it can't distinguish $\mathbb{R}^2$ from $\mathbb{R}^3$: both stay connected after removing a point. To tell those apart, one needs a finer invariant. Removing a point from the plane leaves a hole that a loop can wind around and can't be pulled off; removing a point from $\mathbb{R}^3$ leaves a hole that every loop can slip past. The fundamental group (7A.4 The Fundamental Group) turns that difference into algebra.

4.6Topological spaces, briefly

Tao's optional section defines a topological space: a set $X$ with a collection of subsets, called open, that contains $\varnothing$ and $X$ and is closed under arbitrary unions and finite intersections (2B.1 Metric Spaces). Continuity, compactness (by covers) and connectedness all make sense there, with the definitions of this chapter. What is lost is everything that depends on a metric: Cauchy sequences, completeness, uniform continuity, and (in general) the sequential description of compactness and closure. Book 7A develops topological spaces properly (7A.1 Topological Spaces and Quotients, 7A.2 Compactness and Compactification). Every space in Books 2B to 6A has a metric, so it is safe to skim the section now.

4.7Exercises

Hint

For (a), if $B = U \cup V$ with $U, V$ disjoint and open in $B$, then $A$ lies in one of them, say $U$. Show that $V$ can't contain a point of $\overline{A}$.

Solution

(a) $A = (U \cap A) \cup (V \cap A)$ and $A$ is connected, so $A \subseteq U$, say. If $v \in V$, then $V$ (open in $B$) contains $B \cap B(v, r)$ for some $r$; since $v \in \overline{A}$, this ball meets $A$, so $V \cap A \neq \varnothing$, a contradiction. So $V = \varnothing$. (b) Pick $p \in \overline{A} \cap B$. $A \cup \{p\}$ is connected by (a), and it shares the point $p$ with $B$, so $A \cup B = (A \cup \{p\}) \cup B$ is connected by Proposition 4.9.

Solution

(a) One: the two axes meet at the origin. (b) Four open half-axes. (c) Two branches of the hyperbola, in the first and third quadrants. (d) Four open rays. (e) Each point is its own component: any subset with two points $a < b$ is split by an irrational between them.

Solution

Removing the crossing point of X leaves $4$ components; no point of T leaves more than $3$, and no point of O leaves more than $1$. So X is homeomorphic to neither. Removing the junction of T leaves $3$ components, while removing any point of O leaves $1$. So T and O are not homeomorphic.

Solution

$\det$ is a polynomial in the entries, hence continuous, and on $GL_n$ it never takes the value $0$. If $GL_n$ were connected, its image under $\det$ would be an interval containing $1$ (the identity) and $-1$ (a reflection), hence $0$. So $\{\det > 0\}$ and $\{\det < 0\}$ are disjoint non-empty open sets covering $GL_n$.

Solution

$0 \in S$. $S$ is closed: if $t_n \in S$ and $t_n \to t$, then $|y(s)| \leq 1$ for every $s < t$ (since $s \leq t_n$ for some $n$), and at $s = t$ by continuity. $S$ is open in $[0, T]$: if $t \in S$, then on $[0, t]$ we have $|y'| \leq 1 + 1 = 2$, so $|y(s)| \leq 2s \leq 2t < 1$ by the mean value theorem. Since $|y(t)| < 1$ and $y$ is continuous, $|y| < 1$ on $[t, t + \delta]$ for some $\delta > 0$, so $[0, t + \delta] \cap [0, T] \subseteq S$. As $[0, T]$ is connected, $S = [0, T]$, and the argument just given shows $|y(s)| \leq 2s$ for all $s$.