Reverse mathematics, well-quasi-orders, and Noetherian spaces

A quasi-order $Q$ induces two natural quasi-orders on $P(Q)$, but if $Q$ is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq showed that moving from a well-quasi-order $Q$ to the quasi-orders on $P(Q)$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $P(Q)$ are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form "if $Q$ is a well-quasi-order then a certain topology on (a subset of) $P(Q)$ is Noetherian" in the style of reverse mathematics, proving that these theorems are equivalent to ACA_0 over RCA_0. To state these theorems in RCA_0 we introduce a new framework for dealing with second-countable topological spaces.