Studia Scientiarium Mathematicarum Hungarica 32. (1996)
1-2. szám - Slezák B.: Implicit function theorem in uniform spaces
B. SLEZAK (i) The family of functions T := {/(í,.) 11 G T} is equicontinuous at xo with respect to T iff VWeBy 3V3)x0 VígT: f(t,.)(V)CW(f(t,x0)), so that every partial function f(t,.) is continuous at xq and f(t,.)(V) C C W(f(t,xo)), where V does not depend on t. (ii) The family F is equiopen at Xo with respect to T iff VV3)x 3WeBy VígT: f(t,.)(V)DW(f(t,x0)), so that every partial function f(t,.) is open at xq and f(t, .){V) D W(f(t, xo)), where W does not depend on t. □ Further it is supposed that T is a topological space, X and Y are uniform spaces, Bx denotes a base of the uniformity of X and the elements of By are symmetric. PROPOSITION 2. If the function /(., xo) is continuous at to and the family of functions T := {/(i,.) | tGT} is equicontinuous at xo then f is continuous at (to,xo)Proof. We show that V WeBy 3HB)t0 3V G Bx ■ f(HxV{x0))CW{f(t0,x 0)). Let Wi, W2 G By, W2 o Wy Q W. The neighbourhood H of to can be chosen so that f x0){H) gW2(f {to,x0)), that is (1) VteH: {f{to,xo),f(t,xo))eW2. The set V G Bx can be chosen so that Vi G H: f(t, .)(F(xo)) C Wi(f(t, xo)), hence (2) v(t,x) eH x V{xo): (f{t,xo),f{t,x))eW\. From (1) and (2) it follows that V(f, x) G H x V(x0): (/(i0, ®o), /(*, ®)) eW2oW1gWlgW, hence f(H xV(x0))QW(f (t0, x0)). □ THEOREM 1 (Implicit Function Theorem in uniform spaces). If the function f(.,x0) is continuous at to and T is equiopen at xo then (i) the function F := (prr>/) is open at (to,xo); (ii) for every neighbourhood V(xo) of the point xq there exists a neighbourhood W(f(to,xo)) of f (to, xo) in Y such that for every point of this neighbourhood there is an implicit function belonging to this point, having the fixed set H as a domain and having its range in V(xo), that is