In Stallings's paper
- Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389.
he briefly discusses how to prove "several generalizations" of Brown's theorem saying that monotone union of open $n$-cells is an open $n$-cell; however, he never formulates a precise statement of what he proves.
For the details, he refers to the paper
- J. Stallings, On a theorem of Brown about the union of open cones, Ann. of Math, (to appear).
But as far as I can tell this paper never appeared. Does anyone know what happened and what the theorem is?