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MendelSchmiedekamp comments on Money pumping: the axiomatic approach - Less Wrong
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Sorry I left this out. It's a huge simplification, but treat the set of p as a discrete subset set in the standard topology.
And that is discontinuous; but you can model it by a narrow spike around the value of p, making it continuous.
Hum, this seems to imply that the set of p is a finite set...
Still doesn't change anything about the independence violation, though.
But does doesn't the money pump result for non-independence rely on continuity? Perhaps I missed something there.
(Of note, this is what happens when I try to pull out a few details which are easy to relate and don't send entirely the wrong intuition - can't vouch for accuracy, but at least it seems we can talk about it.)
Actually, I realised you didn't need continuity at all. See the addendum; if you violate independence, you can be weakly money-pumped even without continuity (though the converse may be false).