4. Week 3: Connected Sets
≪ 3. Week 2: Compactness and Continuity | Table of Contents | 5. Week 4: Differentiable Functions ≫Last quarter, you have presumably learned about connected sets, which are an intuitive idea but formalised by the following definition:
Definition 1.
A metric space \(\left( X, d \right)\) is not connected if there exist two nonempty disjoint open sets \(U, V\subseteq X\) such that \(U\cup V = X\). It is connected if it’s not disconnected.
Note that \(U = V^\complement\) and vice-versa, so \(U\) and \(V\) are simultaneously closed and open. Often, one will instead say that \(X\) is connected if the only clopen subsets are \(\varnothing\) and \(X\).
One great application of connectedness is to provide structural support for so-called local-to-global principles. If you have a connected metric space \(X\) and you want to prove that property \(P\) is true everywhere on \(X\), it’s enough to prove that \(P\) is true at a single point in \(X\) and that \(P\) is true on a closed and open subset of \(X\). In fact, the very first example of this is just proving that open path-connected subsets of \(\mathbb{R}^n\) are connected.
Here’s an example:
Theorem 2.
Let \(\left\lbrace U_ \alpha \right\rbrace\) be an arbitrary open cover of a closed interval \(\left[ a, b \right]\subseteq \mathbb{R}\). Then, there exists \(\delta > 0\) such that for all \(a \leq x \leq x+ \delta \leq b\), \(\left[ x, x + \delta \right]\subseteq U_ \alpha \) for some \(\alpha \).
Hint
This idea is sometimes known as the principle of “continuous induction”, and you’ll probably see it elsewhere in pure math (e.g., in complex analysis).
But this begs the question, how does one actually prove that a metric space is continuous to begin with? One generally has three options:
- Use the definition and directly show that the only nonempty clopen subset is the whole metric space itself.
- If \(\left( X, d_X \right)\) is an arbitrary metric space, if \(\left( Y, d_Y \right)\) is connected, and \(f: Y\to X\) is continuous and surjective, then \(X\) is connected. (This is using the fact that the continuous image of a connected set is connected.)
- If \(\left( X, d_X \right)\) is path-connected, then it is connected.
Sometimes, the latter two statements are far easier to prove than the first statement, especially in certain metric spaces where the topology (i.e. open and closed sets) is difficult to think about. But of course sometimes these are not accessible, and one must nonetheless resort to working with clopen subsets anyways.
Problem 3.
Show that the circle \[\mathbb{S} ^1 = \left\lbrace (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \right\rbrace\] is connected with respect to the standard metric on \(\mathbb{R}^2\).
Hint
Problem 4.
Define the “topologist’s comb” in \(\mathbb{R}^2\) as follows: \[C = B_0 \cup \left( \bigcup _{n=1}^{\infty} B_n \right)\cup \left\lbrace (0, 1) \right\rbrace, \] where \(B_0 = \left[ 0,1 \right]\times \left\lbrace 0 \right\rbrace\) and \(B_n = \left\lbrace \frac{1}{n} \right\rbrace \times \left[ 0, 1 \right]\). Show that \(C\) is connected but not path-connected.
Hint
Problem 5. (Basic Exam, Fall 2014 Problem 2)
Let \(A, B\subseteq \mathbb{R}^n\) be two closed subsets such that both \(A\cap B\) and \(A\cup B\) are connected. Show that \(A\) is connected.
Hint
Problem 6. (Basic Exam, Spring 2017 Problem 12)
Let \(K\subseteq \mathbb{R}^n\) be compact. Suppose that for every \(\epsilon > 0\) and pair of points \(a, b\in K\), there exist finitely many points \(x_0,\ldots, x_n\in K \) such that \(x_0 = a\), \(x_n = b\), and \(\left\lVert x_k - x _{k-1} \right\rVert < \epsilon \) for all \(k = 1,\ldots, n\).
- Show that \(K\) is connected.
- Show by example that \(K\) may not be path-connected.
Hint
Problem 7.
Let \(X\) be the set of continuous functions on \(\left[ 0,1 \right]\to \mathbb{R}\), endowed with the uniform metric \[d(f, g) = \sup _{x\in \left[ 0,1 \right]} \left\lvert f(x) - g(x) \right\rvert. \] Show that \(\left( X, d \right)\) is connected.
Problem 8. (Basic Exam, Spring 2020 Problem 9)
Let \(X\) be the set of continuous functions \(f : \left[ 0, 1 \right] \to \mathbb{R}\) such that \(f(1) = 0\) and \(\left\lvert f(x) - f(y) \right\rvert \leq \left\lvert x-y \right\rvert\) for all \(x, y\in \left[ 0,1 \right]\). Define the metric \[d(f, g) = \inf \left\lbrace t \in \left[ 0, 1 \right] : f(x) = g(x) \ \forall t < x \leq 1 \right\rbrace.\] Show that \(X\) is not connected.