Is the space \(\mathbb{R}_\ell\) connected? Justify your answer.
Solution
No. In the lower-limit topology, \[ (-\infty,0)=\bigcup_{n\ge1}[-n,0) \] and \[ [0,\infty)=\bigcup_{n\ge1}[0,n) \] are both open. They are nonempty, disjoint, and their union is \(\mathbb R\). Hence they form a separation of \(\mathbb R_\ell\), so \(\mathbb R_\ell\) is disconnected.