I am reading the paper "Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$ " by Eric Van Douwen. One part of his proof of the Proposition 4 (below) is not clear to me. Suppose $H$ is halfline := $[0; \infty)$.
How does one construct an autohomeomorphism $h$, such that it maps disjoint closed intervals to subsets of neighborhoods of some points in the remainder of compactification of the halfline $H$?
I think such atuhomeomorphism should be linear, at least partly, but I haven´t succeed in explicitly contructing any autohomeomorphism that would match the criteria.
Thank you for any help!
