Homeomorphism from $\mathbb{R}$ onto $[0, \infty)$ with the lower limit topology

Question

I need help with finding a homeomorphism between $\mathbb{R}_l$ (It is, $\mathbb{R}$ with the lower limit topology) and $[0, \infty)_l$ with the subspace topology w.r.t. lower limit topology on $\mathbb R$.

One idea (But I don't know if it works) is to take a function $f$ and each open basis set $[a, b)$ and make $f([a,b))= [e^a, e^b)$ and try to define to extend to $[0, \infty)$ (This continuous function has range $(0, \infty))$

Thanks in advance!

Answer 1

Consider a function $f$ defined in the following way,

$f$ sends $0$ to $0$

$f$ sends ($0,1$] to ($0,1$]

$f$ sends [$-1,0$) to ($1,2$] injectively

$f$ sends ($1,2$] to ($2,3$] injectively

$f$ sends [$-2,-1$) to ($3,4$] injectively

.........

This is injective function but I dont know whether it would be useful for the homeomorphism.