In notes of Susan Niefields about double categories (http://www.tac.mta.ca/tac/volumes/26/26/26-26.pdf) one of the first examples of a double category is a double category structure on $\mathrm{Top}$, where vertical arrows are finite-intersection preserving functions between open set lattices of underlying spaces.
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I struggle to find an explanation for such construction. $\mathrm{Top}$ has more natural structures of double categories, constructed either from its monoidal structure or its bicategory. This one looks clearly not natural, so I assume it is important from some other reasons, for example encodes some properties of spaces. I clearly miss some bigger picture here, so I'll appreciate any insight.