# Japan-United States Seminar On Ordinary Differential And by M. Urabe

By M. Urabe

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Extra info for Japan-United States Seminar On Ordinary Differential And Functional Equations

Example text

J Therefore, E|G and G fl V n c U GtT fl VTT U U satisfies the conclusion of the Proposi tion. 2) Theorem. Let p:E -» B be a locally compact ANR- fibration each fiber of which is a Q-manifold. If B is a countable union of closed sets of finite covering dimension, then p•*E -* B Proof: is a General Position Fibration. , that there exists a Q-manifold fiber preserving maps i:E i >M x B r •E with B, M ri = idp. 1) for some and fix a 32 H. TORUNCZYK AND J. 1) -» M x B. to be an open subset of Furthermore, we may Q.

X } N and observe that are a, then for each disjoint sections of p. U, ( u b ) • extract y •* (^ufn(y)) Since an< p:E -» B is isomorphic with the fibered product of countably many copies of itself, it must be a bundle. Finally, if |A| i> | B | J> K n , then proceed similarly to define a monomorphism a:B -> A and sections f ,,A t g ,, * of *^ a(b) & a ( b ) ^ a(b) A U \ with fa (,, >(b) The Lindelbf Property v v b) ' * &ga ( ,, b )x(b). ' r J of B then allows us to define a countable subset A~(l) of A and disjoint sections of the fibered product of the family {w \a € A n (l)}; repeating this countably many times yields a countable subset A(l) of A such that the fibered product of the ir *s, a € A ( l ) , is a bundle with fiber a Q.

5) provides a fiber- sufficiently close to the projection is proper. Q 1 We define, for = PTTE(f" (f(E x T)) 0 E x S). T € 9 Then with Q-MANIFOLD FIBRATIONS A0 o , Ti (1 U B «j n = 4> and A0 „ i^. i is an F . 31 (By the local compactness o M, f. T €