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

A quasi-order Q induces two natural quasi-orders on P(Q) P(Q) , but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on P(Q) 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) 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) P(Q) is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA0 over RCA0. To state these theorems in RCA0 we introduce a new framework for dealing with second-countable topological spaces.