Problem [Munkres, 9, page 171] Problem. Generalize the tube lemma as follows: Let A and B be subspaces of X and Y , respectively; let N be an open set in X Y containing A B. If A and B are compact, then there exist open sets U and V in X and Y , respectively, such that A B U V N. Solution: We are going to adapt the proof of the tube lemma to this situation. Let a A. Using the compactness of {a} B, cover {a} B with nitely many products U1 V2 , U2 V2 , . . . , Un Vn so that all Ui and Vi are open and {a} B Ui Vi N. I f some of the Ui Vi do not intersect {a} discard B, them. Setting U = U1 U2 Un and V = V1 V2 Vn , we have {a} B U V N. To see this, we rst note that a U . Indeed, since Ui Vi intersects {a} B, we have a Ui for all i, and a U . Also, if b B, then for some i, a b Ui Vi which implies a b U Vi U V. Thus {a} B U V . Finally, U V =U ( Vi ) = (U Vi ) (Ui...