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Example text

Let f: X -+ Y be a OP-morphism. Let x ~ E 1) modelled on X. Then: (i) f is an immersion at x if and only if there exists a chart (U, lp) at x and (V, 1/1) at f(x) such that fv,u(lpx) is injective and splits. (ii) f is a submersion at x if and only if there exists a chart (U, lp) at x and (V, 1/1) at f(x) such that fv,u(lpx) is surjective and its kernel splits. Proof. This is an immediate consequence of Corollaries 1 and 2 of the inverse function theorem. The conditions expressed in (i) and (ii) depend only on the derivative, and if they hold for one choice of charts (U, lp) and (V, 1/1) respectively, then they hold for every choice of such charts.

Let w(x) = cp(x) (1 - CPi(X)). n Then w(x) satisfies our requirements. Theorem 2. Let A l , A2 be non-void, closed, disjoint suhsets of a separable Hi7JJert space E. Then there exists a Goo-function t/I: E - R suck that t/I(x) = 0 if x E Al and t/I(x) = 1 if x E A2, and 0 ~ t/I(x) ~ 1 for all x. Proof. By Lindelof's theorem, we can find a countable collection of open balls rUt} (i = 1,2, ... ) covering A2 and such that each Ut is contained in the complement of A l . Let W be the union of the Ui.

We construct inductively a sequence A 1 , A 2 , ••• of com- pact sets whose union is X, such that At is contained in the interior of At+1' We let A1 = U1. Suppose we have constructed A,. We let j be the smallest integer such that At is contained in U1 U··· U Uj. We let A'+1 be the closed and compact set U1 u ... +1' For each point x E X we can find an arbitrarily small chart (V x, lpx) at x such that lpx V x is the ball of radius 3 (so that each V x is contained in some element of U). We let W x = lp;1(B1) be the ball of radius 1 in this chart.