Commentationes Mathematicae by Orlicz L.

By Orlicz L.

The current quantity of Commentationes Mathematicae is devoted to its Editor, Professor Wladyslaw Orlicz, at the social gathering of his 75-th birthday.Professor Wladyslaw Orlicz, a very good mathematician, used to be as a member of the recognized Lwow university one of the founders of contemporary sensible research. The articles accumulated during this quantity, lots of them via Professor Orlicz's former scholars, have been encouraged by means of the guidelines raised within the paintings of Professor Orlicz. it's, we think, the simplest image of recognize that his pals worldwide carry for Professor Orlicz.

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For each xeC, let Gx be open and such that xeGx and Gx is compact. ,xk be a finite set of points such that C c G = GXl u ... u Gx . Clearly G is compact and &~2 (compare II, Theorem 5), hence normal (by II, Theorem 3). Consequently there is a continuous g: G -> J such that g(x) = 0 for xeG and g(x) = 1 for xe[(G n F) u (G-G)]. Put f(x) = g(x) for xeG and f(x) = 1 for xjG. Clearly / is continuous and satisfies condition (1). Finally, if f(x)

Since Kx<& is closed in ^ χ ^ , the following set is closed E\(Q(y)nD)n(Kx&) Φ0]. [F(y) nK φθ]. ïfow this follows from the formulas: y ((x,y)cD)(xeK) n(ixf)] =(x,y)c[D = {n,y)*lQ(y) hence [F(y) nK ^0] s X πΰπ(ίχ^)], VhNni)|HV[(^) X = [Q(y)nDn(Kx&) THEOREM 4. Let F: &->2^. only if the set §WnDn(ixf)] ^0]. c. if and D = Ei*eF(y)i xy e (2) is closed. Proof. c, then D is closed (by Theorem 1 of § 18, III, which is valid for any regular X). Conversely, if D is closed, our statement follows from Theorem 3 since {x,y)eD ~x€F(y).

Then X is locally compact if and only if 2X is such (2). Π. Case of X compact metric. In § 21, VII, we have denoted, for each bounded metric space X, by {2x)m the metric space of all closed subsets of X, the (Hausdorff) distance, dist(JL,J5), of two sets A Φ 0 and B Φ 0 being defined as the lower upper bound of the numbers ρ(χ,Β) and q{y,A) where xeA and yeB; the void set has been defined as an isolated point of (2^) m . THEOREM. Let X be compact metric. Then 2\=(2*)m. Namely, the identity mapping: 2X -> {23C)m is a homeomorphism onto.

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