What are the components and path components of \(\mathbb{R}_\ell\)? What are the continuous maps \(f: \mathbb{R} \to \mathbb{R}_\ell\)?
Solution
1. The components and path components of \(\mathbb R_\ell\) are the one-point sets.
For \(a\in\mathbb R\), the rays \((-\infty,a)=\bigcup_{n\ge1}[a-n,a)\) and \([a,\infty)=\bigcup_{n\ge0}[a+n,a+n+1)\) are open in \(\mathbb R_\ell\), disjoint, and cover \(\mathbb R\). If \(x<y\), choosing \(a\) with \(x<a\le y\) gives a separation of \(\mathbb R_\ell\) with \(x\) and \(y\) on different sides, so no connected set contains both. Hence the components are points, and so are the path components, each of which lies in a component.
2. The continuous maps \(f\colon\mathbb R\to\mathbb R_\ell\) are exactly the constant maps.
Constant maps are continuous. If \(f\) is continuous, \(f(\mathbb R)\) is connected because \(\mathbb R\) is, so by step 1 it is a single point.
3. Q.E.D.
Steps 1 and 2 answer the two questions.