Find a topological space $X$ which is connected but has three path components.

Question

What would a connected topological space $X$ look like with three path components? I know that since it has a finite number of path components, these components are closed but I'm not sure if that helps.

Answer 1

One can find examples as subspaces of $\mathbb{R}^n$. Try taking advantage of the fact that any set between (in the sense of containment) a connected set and its closure—up to the closure itself—is also connected*. It can be the case, when $n>1$, that the closure of a path-connected (hence connected) set $X$ is no longer path-connected, which means we can potentially increase the total number of path components by passing from a connected set to its closure. Subsequently, there's a chance we can get even more by deleting points from path components in $\overline{X} \setminus X$, all the while preserving connectedness.

For instance, consider the path-connected set $X = \left\{ \big(x, \sin(1/x) \big) \ | \ x \in (0, 1] \right\} \subset \mathbb{R}^2$ (cf. the topologist's sine curve). What is $\overline{X}$? You'll find that $\overline{X}$ is no longer path-connected, having two path components. Moreover, by deleting points from $\overline{X} \setminus X$, we can construct a set $Y$, where $X \subset Y \subset \overline{X}$, such that $Y$ has arbitrarily many path components (even as many as $\aleph_0$ or $\aleph_1$) while still being connected.

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*For proof, see Munkres' Topology: Chapter $3$, theorem $23.4$

Answer 2

The following is easy to prove:

Proposition 1: Let $X$ be any topological space with a nested decreasing chain of nonempty open subsets:

$\tag 1 U_0 \supset U_1 \supset \dots \supset U_n \supset \dots$

Let $\rho$ be any element not in $X$. Then the collection of sets $\mathcal B$ defined by

$\tag 2 U \in \mathcal B \text{ iff [} U \text{ is open in } X \text{ or } U = U_n \cup \{\rho\} \text{]}$

forms a topological basis for the set $\hat X = X \cup \{\rho\}$.

For the topological space $\mathbb R$ we can define the decreasing chain of open sets

$\tag 3 U_n = \mathbb R - \mathbb Z - \{x \in \mathbb R \, | \; |x| \lt n \}$

and can therefore create the space $\hat {\mathbb R}$ as shown above, which is also a Hausdorff space.

Proposition 2: The space $\hat {\mathbb R}$ is connected with two path components.
Proof
It is easy to argue that any clopen containing $\rho$ is all of $\hat{\mathbb X}$, so we have a connected space.

We show that if $\gamma$ is any path with $\gamma(0) \in \mathbb R$, then the image $\gamma([0,1])$ is also contained in $\mathbb R$. Recall that the image any path is connected.

If not, there is a map $\gamma^{'}$ with $\gamma^{'}([0,1)) \subset (n, n+1)$ and $\gamma^{'}(1) = \rho$. But for continuous functions the image of the closure is contained in closure of the image, yet $\rho \notin [n, n+1]$. $\qquad \blacksquare$

Let there be given two disjoint copies $\mathbb R_1$ and $\mathbb R_2$ of the number line and take $\mathbb X$ to be the disjoint sum of these spaces. We will now define a decreasing chain ${\mathbb U}_n$ of open sets in $\mathbb X$. For $i \in \{1,2\}$, set

$\tag 4 {U_n}^i = \mathbb R_i - \mathbb Z_i - \{x \in \mathbb R_i \, | \; |x| \lt n \}$

and

$\tag 5 {\mathbb U}_n = {U_n}^1 \sqcup {U_n}^2$

So by proposition 1 we have a space $\hat{\mathbb X}$.

Proposition 3: The space $\hat{\mathbb X}$ is connected with three path components.
Proof
Let a path $\gamma: [0,1] \to \hat{\mathbb X}$ be given, and suppose that, say, $\gamma(0) \in \mathbb R_1$. By continuity, there is an $a \gt 0$ such that the image of the interval $[0,a)$ under $\gamma$ is contained in $\mathbb R_1$. If $\gamma$ maps any points outside of $\mathbb R_1$ then there is a map $\gamma^{'}$ with $\gamma^{'}([0,1))$ contained in $\mathbb R_1$ and $\gamma^{'}(1) \in \mathbb R_2 \cup \{\rho\}$. But then in either case we can get a contradiction. $\qquad \blacksquare$

This argument can be extended to show that we can construct a connected space with an arbitrary number of path components (Kaj Hansen also points this out). In this construction exactly one path component is a singleton set.