The answer is yes. The implications 1$\iff$3 follow because $[0,\infty]$ is homeomorphic to $[0,1]$. The proof of the implications 2$\iff$3 is almost the same as in the case of $[0,\infty)$ or $(-\infty,\infty)$ in place of $[0,\infty]$, if $\infty+\epsilon:=\infty$ for real $\epsilon$.
Iosif Pinelis
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