$[0,\infty)_{l}$ and $R_{l}$ are homeomorphics

Question

Let $[0,\infty)_l$ the subset $[0,\infty)\subset R_{l}$ with the subspace topology.

I need to prove that $R_l$ and $[0,\infty)_l$ are homeomorphics.

We can see that the continuos functions of $R$ to $R_{l}$ are continuous from the right. Then I tried with a function by parts with all parts nondecreasing, but this function don't send opens sets on opens sets $[0,\infty)_l$ to $R_{l}$. Please, can you give me some ideas?

Answer 1

Write $[0,\infty) = \bigcup_{n=0}^\infty [n,n+1)$ and $\mathbb{R} = \bigcup_{n \in \mathbb{Z}} [n,n+1)$ which is a partition of both spaces into trivially homeomorphic (just a translation will do) clopen (in the lower limit topology) parts.

Now use your favourite bijection $b: \mathbb{Z} \to \{0,1,2,3,\ldots\}$ to define the homeomorphism.