A Four Flagged Lemma

Cover A Four Flagged Lemma
A Four Flagged Lemma
Murat R Sertel
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= ir (x) . Then any function f which makes the 1 X^ diagram -> v(X) commute is continuous, -3- Proof ; Let W be a subbaslc open set in E^. It suffices to show that f~^W is open. Assume, without loss of generality, that W = {w e E^ I w > w }. If w i u(X), then the connectedness of u(X) C E^ implies that either (i) w u(x) for all x e X. If (i) holds, then u"^(W) = X, so that v(u"nw)) = v(X) = f~^(W) is open. If (ii) holds, then u"^(W) = = v(u"^(W)) = f"^(W) is open. So, assume w* e u(X), * * * i.... E. , that w = u(x ) for some x e X. Then, as u preserves the complete preorder x }, which is open (showing u to be continuous) by the semiclosedness of v^(x ^)}, where x ^ = tt^ (x ), so that i v^(Pj, ) is again open. Thus v(u-^(W)) = H v^(P^) is open, and, M from the commutativity v u"^ = f"^, we conclude that f~'^(W) is open, as to be shown.

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