This article was originally intended to be about a notion of connectedness — chain-connectedness — which I find especially intuitive and visually appealing. In the process of writing, I realized that in practice, chain-connectedness isn’t always the most efficient or appealing proof method. Often, it is easier to prove that some property holds locally, in a neighborhood of each point, than to prove it holds globally, for an entire space. Chain-connectedness offers a visual interpretation for how such local properties are patched together and extended to global ones, but it does not explicitly characterize which local properties can be extended to global ones.

In what follows, I describe my three favorite characterizations of connectedness and explain how they are related. The first frames connectedness as a type of structural induction, the explanation of which I delegate to another blog; the second is chain-connectedness, a discretization of path-connectedness; the final characterization makes precise which local properties can be extended to global ones and frames connectedness as the principle that any distinction of points which vanishes locally must vanish globally.

Connectedness as an Induction Principle

Our initial characterization of connectedness will be as follows:

Definition (Connected)

A topological space $X$ is said to be disconnected provided that there exists two non-empty, disjoint, open subsets of $X$ which union to $X$. The space $X$ is said to be connected provided that it is not disconnected.

Intuitively, this notion of connectedness says that the non-empty open subsets of $X$ cannot separate $X$.

Over at Pseudonium’s Blog, connectedness as an induction principle is motivated by and generalizes real induction. Roughly, the induction principle characterizes a connected space by its clopen subsets, of which there can only be two — the whole space and the empty space.

Theorem (Induction Principle for Connectedness)

A topological space $X$ is connected if and only if for every subset $S$ of $X$ such that

• $S$ is non-empty
• $S$ is open
• $S$ is closed

then $S = X$.

Proof

Suppose that $X$ is connected. Let $S$ be a subset of $X$ satisfying the three properties above, i.e., $S$ is a non-empty clopen subset. If $S$ is a proper subset of $X$, then $X - S$ is also a non-empty, open subset of $X$; $X$ is thus the union of two non-empty, disjoint, open subsets, a contradiction.

Conversely, suppose $X$ is a space satisfying the inductive principle — that the only non-empty clopen subset of $X$ is $X$. Let $U$ and $V$ be two non-empty, disjoint, open subsets which union to $X$. The set $U = X - V$ is a non-empty, clopen subset of $X$. By the inductive principle on $X$, $U = X$, contradicting the fact that $V$ is non-empty. Since $X$ cannot be disconnected, $X$ is connected.

This characterization describes necessary and sufficient conditions that a subset of a connected space must satisfy to be the entire space. If a subset $S$ of a connected space $X$ is defined by a predicate $\phi$ on $X$ as $S = \{x \in X : \phi(x)\}$, the induction principle for connectedness states that $\phi$ is true at every point of $X$ provided that

• $\phi(x)$ is true at some point $x \in S$,
• on $S$, $\phi$ is locally true, meaning that for each inhabitant $x \in S$, there is an open set $U$ such that $x \in U \subseteq S$ and $\phi(u)$ is true for all $u \in U$, and
• on $X - S$, $\phi$ is locally false.

We have met the first of my three favorite perspectives on connectedness.

It is hard not to mention that this last observation leads naturally to the characterization of connectedness by continuous functions into discrete spaces; however, this isn’t one of the three perspectives I will be discussing.

Related is the notion of compactness as an induction principle.

Chain-Connectedness

After learning about chain-connectedness during my undergrad (my original favorite characterization), I found it to be extremely intuitive, easy to visualize, and made certain proofs more transparent. Additionally, like its topological counterpart — compactness — chain-connectedness is a characterization of connectedness in terms of open covers.

Definitions

The preliminary notion of a chain is useful for defining chain-connectedness:

Definition (Chain and Chain-Connected Open Covers)

Let $X$ be a topological space and $\mathcal{U}$ be an open cover of $X$ (a collection of open subsets of $X$ which union to $X$).

Given two open subsets $U,V \in \mathcal{U}$, a chain from $U$ to $V$ in $\mathcal{U}$ is a finite sequnece $\{U_0,\dots,U_{n}\}$ in $\mathcal{U}$ such that

• $U_0 = U$,
• $U_n = V$, and
• $U_{i} \cap U_{i + 1} \neq \emptyset$ for each $0 \leq i < n$.

For $x \in U$ and $y \in V$, the sequence $\{U_{0},\dots,U_{n}\}$ is called a chain from $x$ to $y$ in $\mathcal{U}$.

The open cover $\mathcal{U}$ is a chain-connected open cover if there is a chain between any two non-empty open sets in $\mathcal{U}$, or equivalently, between any two points of $X$.

As the name suggests, the open sets of a chain (in an open cover) form the chain links, and they are linked together by having non-trivial overlap. Just like jewelry, chains can loop back on themselves and have repeated links.

Definition (Chain-Connected)

A topological space $X$ is said to be chain-connected if every open cover of $X$ is a chain-connected open cover of $X$.

Explicitly, $X$ is chain-connected if for every open cover $\mathcal{U}$ of $X$ and any two non-empty open sets $U,V \in \mathcal{U}$, there is a finite sequence $\{U_{0},\dots,U_{n}\}$ in $\mathcal{U}$ such that

• $U_{0} = U$,
• $U_{n} = V$, and
• $U_{i} \cap U_{i + 1} \neq \emptyset$ for each $0 \leq i < n$.

Before moving on, let’s contrast this with the notion of path-connectedness:

Definition (Path and Path-Connected)

A path in a topological space $X$ is a continuous function $\gamma : I \to X$, where $I = [0,1]$ is the unit interval (with the usual topology).

The space $X$ is said to be path-connected provided that there is a path between any two points of $X$.

The notion of chain-connectedness may be seen as a discretization of path-connectedness; instead of any two points being joined by a path of points in $X$, roughly, any two open sets (or equivalently, points) are joined by a path of open sets in $X$. This observation suggests a natural visualization of chains and chain-connectedness.

The main reason I like chain-connectedness so much is because of its visual and geometric nature which I feel other characterizations lack; it becomes more intuitive after trying to apply it, as we will now.

The Link to Connectedness

It is a familiar theorem that every path-connected space is connected. The discrete analogue of this is also true; every chain-connected space is connected. As suggested, these notions of connectedness are also equivalent.

Theorem (Chain-Connectedness and Connectedness are Equivalent)

A topological space $X$ is connected if and only it is chain-connected.

Proof

Apply the induction principle for connectedness!

Let $\mathcal{U}$ be an open cover of $X$. Declare two points $x$ and $y$ of $X$ to be equivalent if and only if there is a chain in $\mathcal{U}$ from $x$ to $y$ and write $x \sim y$; indeed, this defines an equivalence relation on $X$. This equivalence relation will be referred to as the equivalence relation determined by chains.

Say that $x_0$ is a point of $X$; define $S = \{x \in X : x \sim x_0\}$.

• $S$ is non-empty: $x_0 \in S$.
• $S$ is open: for $x \in S$, there is a chain $\{U_0,\dots,U_n\}$ in $\mathcal{U}$ from $x_0$ to $x$. For each $y \in U_n$, the same chain is a chain from $x_0$ to $y$, thus $U_n$ is an open neighborhood of $x$ contained in $S$.
• $S$ is closed: for $x \in X - S$, there is an open set $U \in \mathcal{U}$ containing $x$. $U$ cannot intersect $S$, since otherwise there would be a chain from $x_0$ to $x$.

By the inductive principle for connectedness, $S = X$, thus $X$ is chain-connected.

Conversely, suppose $X$ is chain-connected. Let $S$ be a non-empty clopen subset of $X$. The set $\mathcal{U} = \{S, X - S\}$ is an open cover of $X$, so there is a chain in $\mathcal{U}$ between any two of its non-empty open subsets. No finite sequence in $\mathcal{U}$ can form a chain from $S$ to $X - S$; therefore $X - S$ must be empty, i.e., $S = X$. Having demonstrated that $X$ satisfies the induction principle for connectedness, $X$ is connected.

Having shown that these two notions of connectedness are equivalent, we have met the second of my three favorite perspectives on connectedness.

You can find chain-connectedness briefly discussed in General Topology by Willard, and as exercises in General Topology by Engelking and Topology by Munkres.

Connectedness as a Local-to-Global Principle

Any subset $S$ of a set $X$ induces an equivalence relation $x\sim y$ if and only if $x,y \in S$ or $x,y \in X - S$. If $X$ is a topological space, and $S$ is clopen, then the equivalence classes of $\sim$, $S$ and $X - S$, are both open. If we think of equivalence relations as a way to distinguish points of a space, the distinction of points of $X$ by belonging to $S$ or not belonging to $S$ vanishes locally. In a connected space, that is enough for the distinction to vanish globally; therefore, if $X$ is connected, and $S$ is non-empty, then $S = X$.

This observation leads us to my final favorite perspective on connectedness. At least one other source calls this the local-to-global lemma. First, it is useful to make precise what is meant by a distinction vanishing locally and globally.

Definition ((Locally) Trivial Equivalence Relation)

An equivalence relation $\sim$ on a set $X$ is trivial if $\sim \, = X \times X$, that is, if every element of $X$ is related to every other element of $X$, or, if $\sim$ has a single equivalence class.

If $\sim$ is an equivalence relation on a topological space $X$, $\sim$ is called locally trivial if there is an open neighborhood $U$ around each point of $X$ on which $x \sim y$ for all $x,y \in U$ (equivalently, if the equivalence classes of $\sim$ are open).

Locally trivial equivalence relations are ways to distinguish points of a space where the distinction vanishes locally, i.e., each point is contained in an open neighborhood where all points are “equal”.

Theorem (The Local-to-Global Lemma)

A topological space $X$ is connected if and only if every locally trivial equivalence relation on $X$ is (globally) trivial.

Proof

Suppose $X$ is connected. Let $\sim$ be a locally trivial equivalence relation on $X$. Each equivalence class of $\sim$ is non-empty and clopen. By the induction principle for connectedness, each equivalence class is equal to $X$. The equivalence relation $\sim$ has a unique equivalence class and is therefore trivial.

Conversely, let $S$ be a non-empty clopen subset of $X$. The equivalence relation $x \sim y$ if and only if $x,y \in S$ or $x,y \in X - S$ is a locally trivial equivalence relation, and is thus promoted to a globally trivial one. Let $x_0 \in S$. Every element $x \in X$ is related to $x_0$, and so $S = X$. By the induction principle for connectedness, $X$ is connected.

The local-to-global lemma frames connectedness as the principle that any distinction which vanishes locally must vanish globally. It also says that the local properties which can be extended to global ones are exactly those which induce a locally trivial equivalence relation.

It is interesting to specialize the above proof of the converse direction to the case when $S$ is defined by a predicate on $X$. It is also interesting to specialize the above proof of the forwards direction when $\sim$ is the equivalence relation determined by chains.

Applications

Here are a few applications of my favorite notions of connectedness.

Proposition

If $X$ is connected and locally path-connected, then $X$ is path-connected.

Proof

Path-connectedness is an equivalence relation on any space; being locally path-connected means that it is a locally trivial equivalence relation (locally, any two points are equivalent). By the local-to-global lemma, $X$ is path-connected.

Proposition

Every locally constant function on a connected space $X$ is constant.

Proof

Let $f : X \to Y$ be a locally constant function on a connected space $X$.

Say that $x \sim y$ if and only if $f(x) = f(y)$. Any distinctions among points with respect to $\sim$ vanishes locally ($f$ is locally constant); by the local-to-global lemma, $f$ is constant.

Proposition

If $A$ is a connected subspace of $X$, if $A \subseteq B \subseteq \mathtt{cl}\,A$, then $B$ is connected.

Proof

We use chain-connectedness.

Let $\mathcal{U}$ be an open cover of $B$ and fix $x,y \in B$. There are four cases:

1. $(x \in A, y \in A)$ There is a chain in $\mathcal{U}$ from $x$ to $y$ since $A$ is connected.

2. $(x \in A, y \in B - A)$ In this case, $y$ is a limit point of $A$; there is some neighborhood $U \in \mathcal{U}$ of $y$ and some $z \in U \cap (A - \{y\})$. By (chain-)connectedness of $A$, there is a chain in $\mathcal{U}$ from $x$ to $z$. Appending $U$ to this chain produces a chain in $\mathcal{U}$ from $x$ to $y$.

3. $(x \in B - A, y \in A)$ This case is symmetric.

4. $(x \in B - A, y \in B - A)$ Just as in case 2., there is an open set $U_y \in \mathcal{U}$ containing $y$ and some $z_y \in U_y \cap (A - \{y\})$; likewise for $x$. By (chain-)connectedness of $A$, is a chain in $\mathcal{U}$ from $z_x$ to $z_y$. Prepending $U_x$ and appending $U_y$ yields a chain from $x$ to $y$ in $\mathcal{U}$.

The first two propositions above can also be proven using chain-connectedness, which shows more explicitly how to patch together the local data into global data.

This next batch of applications uses a bit more machinery, so I will only give proof sketches.

Here is an adaptation of Problem 4-31 from Lee’s Introduction to Topological Manifolds.

Proposition

Let $X$ be locally Euclidean Hausdorff of dimension $n$ and paracompact.

If $X$ has countably many connected components, then $X$ is a topological $n$-manifold, that is, a second countable locally Euclidean Hausdorff space.

Proof

It suffices to assume that $X$ is connected — such a space is locally connected, so its connected components are open. A countable basis on each connected component of $X$ therefore assembles into a countable basis on the entire space.

Let $\mathcal{V}$ be the collection of all open subsets of $X$ which are $n$-manifolds and whose closure is compact; $\mathcal{V}$ is non-empty and covers $X$ since each point is contained in such a neighborhood. By paracompactness, there is a locally finite refinement $\mathcal{U}$ of $\mathcal{V}$ whose elements are all non-empty.

Fix $U_0 \in \mathcal{U}$; by (chain-)connectedness, there is a finite chain in $\mathcal{U}$ from $U_0$ to $U$ for $U \in \mathcal{U}$. Let $f : \mathcal{U} \to \mathbb{N}$ be the function defined by taking each $U \in \mathcal{U}$ to the length of a shortest chain from $U_0$ to $U$ in $\mathcal{U}$. By local-finiteness, the fibers of $f$ are finite, thus $\mathcal{U}$ is countable.

Any locally Euclidean Hausdorff space of dimension $n$ which can be covered by countably many compact sets is second countable (see Problem 4-16 or this paper for details).

For each point $p$ of the open unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$, there is a smooth diffeomorphism of $\mathbb{R}^n$ which is the identity outside of $\mathbb{B}^n$ and takes the origin to $p$. Such a map can be chosen to be smoothly isotopic to the identity — a description can be found in Milnor’s Topology from a Differentiable Viewpoint on pages 23 and 24.

By solving the problem locally, we obtain the following homogeneity result:

Proposition

Let $M$ be a connected smooth manifold. The diffeomorphism group of $M$ acts transitively on $M$. Moreover, between any two points, one can choose a diffeomorphism between them which is smoothly isotopic to the identity.

Proof

Say two points $x,y$ of $M$ are equivalent if there is diffeomorphism which is smoothly isotopic to the identity and takes $x$ to $y$. This determines a locally trivial equivalence relation. By the local-to-global lemma, any two points of $M$ are related by a diffeomorphism.

A closely related result is (an adaptation of) Problem 5-4 from Lee’s Introduction to Topological Manifolds:

Proposition

Let $M$ be a connected manifold of dimension $2$ or greater. Let $(p_1,\dots,p_k)$ and $(q_1,\dots,q_k)$ be ordered $k$-tuples of points in $M$. There is a diffeomorphism $F : M \to M$ such that $F(p_i) = q_i$ for each $i = 1,\dots,k$.

Proof

Following the sketch outlined here, the equivalence relation described is a locally trivial one, so one can repeatedly apply the local-to-global lemma to obtain the result.

The last two results results also hold in settings of varying degrees of smoothness by replacing diffeomorphism with $C^k$-diffeomorphism or homeomorphism and weakening or removing the smoothness requirements.

Closing Thoughts

While these notions of connectedness are my favorite three, and together they helped me deepen my intuition for connectedness, they are not the only characterizations of connectedness that one should focus on; each characterization offers a different perspective and may be better suited to a certain problem. For example, none of the three characterizations above give efficient proofs that the product of connected spaces is connected (even in the case of finitely many factors) as the characterization of connectedness by continuous maps into $\{0,1\}$ is better suited to prove this (at least in the case of finitely many factors).

Let me know if you can think of any more applications of these characterizations of connectedness!