$\mathbb{R}$ is not homeomorphic to $[0,\infty)$

1. Suppose that a homeomorphism\[ f:[0,\infty)\longrightarrow\mathbb R \]exists, and set \(y=f(0)\).

2. Restricting \(f\) gives a homeomorphism\[ (0,\infty)=[0,\infty)\setminus\{0\} \longrightarrow \mathbb R\setminus\{y\}. \]

3. The space \((0,\infty)\) is connected, whereas\[ \mathbb R\setminus\{y\}=(-\infty,y)\sqcup(y,\infty) \]is disconnected.

4. This contradicts invariance of connectedness under homeomorphism. Therefore \(\mathbb R\) is not homeomorphic to \([0,\infty)\).