f(x) = f-1(x) (*) 343

RETRACTING MULTIFUNCTIONS BY GORDON T. WHYBUIIN UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE, VIRGINIA Communicated December 12, 1967 (1) Nonmingled Multifunctions.-If X and Y are sets, a multifunction (setvalued) f: X -- Y is nonmingled provided the images of any two points in X are either disjoint or identical; that is, x1,x2 e X gives either f(x,) f(x2) = c or f(x1) f(x2). It is not difficult to verify that such a multifunction f is nonmingled if and only if any one of the following conditions is satisfied: (a) ff-1f fonx. (b) f-f-1 - f-1 on f(x). (c) For each image set Bcf(X), ff-1(b) = B. (d) For each inverse set A CX, f-lf(a) = A. (e) For any inverse set AcX, f(x - A) = f(x) - f(a). For example, to show that (c) implies nonmingledness, let Xl,X2 e X and choose B = f(x1). Then if f(x2) meets B, x2 e f-1(b); and we have f(x2)cb = f(xi). Similarly f(xi) cf(x2). On the other hand, suppose f is nonmingled and take any image set B in f(x), say B = f(a) for ACX. If, contrary to (c), there existed a point X2 in f-i(b) with f(x2) not a subset of B, we could take a point xl in A with f(x1) containing a point of f(x2) * B. Since f(x1) cf(a) = B, this gives f(xl) X f(x2) contrary to nonmingledness. We note also that from (a) and (b) it results at once that if f is nonmingled, so also is f-1. We remark further that the nonmingled condition is not transitive for multifunctions. It does not follow that if f: X =) Y and g: Y-=) Z are nonmingled, then gf: X ==) Z is nonmingled. (The double arrow ==) means that the relation is onto.) For let f map a pair of distinct points into disjoint intervals A1 and 12. Then let g identify one end of I, with one end of I2. In this case gf is mingled even though g is single-valued and continuous. However, if f (as above) is single-valued and g is nonmingled, then gf is necessarily nonmingled. (2) Retracting Multifunctions.-If YCX a multifunction, f: X -0 Y is said to be retracting provided that y e f(y) for each y e Y. It results from this definition, of course, that all retracting multifunctions are onto. Also it is seen at once that f is retracting if and only if y e f-1(y) for each y e Y. Thus in particular in case X = Y, f-1 is retracting if and only if f is retracting. Iff: X Y is retracting, so also is fix0 for any subset Xo of X. In particular fj Y is retracting if and only if f is retracting. Thus if ft Y is retracting, so also is fix no matter how f be defined on X - Y. In view of 'this fact it is clear that we may as well limit our study to the case in which Y = X, since no additional interest is inherent in the more general situation. Thus we consider only multifunctions of the type f: X =) X. (2.1) If f: X ) X is nonmingled and retracting, then for each x E X we have f(x) = f-1(x) (*) 343

344 MATHEMATICS: G. T. WHYBURN PRoc. N. A. S. Proof: For any x' ef-1(x) we have f(x') Dx + x'. Thus x' e f(x') = f(x), sincef is nonmingled. Whencef-1(x)cf(x). On the other hand, x Ef(x) gives Af(x') = f(x) so that x' ef-'(x). Thusf-'(x) Df(x). This proves (*). Remark: The converse of (2.1) does not hold. The involutionf(x) = -x on the real axis is nonmingled and satisfies (*) but is not retracting. Other relevant examples are given below. (2.2) A nonmingled multifunction f: X ==) X is retracting if and only if for each x ex, f(x) = ff(x). (t) Proof: Suppose (t) holds and take any y e X. Find x e X so that y e f(x). Then f(y) cff(x) = f(x). Whence, f(y) = f(x) since f is nonmingled. Accordingly, y e f(y). On the other hand, suppose f is retracting and take any x e X. Then ff(x) D f(x) since f is retracting. Suppose contrary to ff(x) cf(x) that for some y ef(x), f(y) is not contained in f(x). Then f(y) # f(x) whereas y e f(x).f(y); and this is impossible because f is nonmingled. Examples: For mingled multifunctions neither of these implications necessarily holds. For if we define f(x) = (x, co) (the open interval from x to co), for x on (0, ac), then f(x) = ff(x) for all x but f is not retracting. Again if we define f(x) = [x,2x] for x e [0, co), thenf is retracting but ff(x) $ f(x) for all x > 0. (2.3) THEOREM. A multifunctionff: X=)Xisnonmingled and retracting if and only if both (*) and (t) hold for each x e X, i.e., f(x) = ff(x) = f-1(x). (;) Proof: If f is nonmingled and retracting, (*) holds by (2.1) and (2.2). Thus we suppose (*) holds. Then to complete our proof it suffices, in view of (2.2), to show that f is nonmingled. To that end we take any x, y e X. Then if y e f(x), we have (i) f(y) cff(x) = f(x). Also (ii) x e f1(y) = f(y). Thus (ii) and (i) give f(x) cf(y) so that f(x) = f(y). Hence y e f(x) implies f(y) = f(x). Now suppose that for some x, y e X, there is a point z e f(x) -f(y). By what was just shown we have f(z) = f(x) = f(y). Hence f is nonmingled. Examples: Nonmingledness does not follow from either (*) or (t) alone even though f is retracting. For let A be any triangle and define f(x) to be the union of all sides of A containing x for x e A. Then (*) holds and f is retracting but mingled. On the other hand, for x e [0,1 ] if we define f(x) = [x, 1 ], (t) holds and f is retracting but mingled. As an easy consequence of (2.4) together with the fact noted earlier that f-1 is retracting if and only if f is retracting and likewise f-1 is nonmingled if and only if f is nonmingled, we obtain the following result. (2.5) If f: X =) X is nonmingled and retracting, then for any image or inverse set I, we have f(i) = I = f-, V) - (*)

VOL..59, 1968 MATHEMA TICS: G. T. WHYBURN 345 (3) Decompositions and Retractions.-It is now clear that a retracting multifunction ff: X =) X simply defines the decomposition of X into the sets [f(x) ], x e X. Thus such a multifunction could be called the decomposing function for a given decomposition. In case f has nonmingled images, the decomposition G,generated is into disjoint sets and conversely. Assuming X to be a topological space, we now consider the question: How is the semicontinuity of f related to that of G? We recall that a decomposition G of X is upper (lower) semicontinuous, written use (lsc), provided that the union of all elements of G intersecting any closed (open) set is closed (open). Also, a multifunction f: X -) Y is upper (lower) semicontinuous if and only if for each closed (open) set Cc Y, f-1(c) is closed (open). Let G be any decomposition of X into disjoint sets. Define the multifunction 0D: X ==) X by 4,(x) = the element of G containing x. Then 4, is retracting and noniningled and is called the natural retraction of (or generated by) G. For CcX, let K be the union of all g e G intersecting C. Then since x e ckg-'(c) SC K is means 4,(x) -C X A, we have ck-'(c) = K. Accordingly, if G is (closedn h C {closedn that.lsci Conversely, y if 5, is ( open J open l JDSC +,-1(C) is closed when C is closed so that G is SC Now let us reverse the process. Let 0: X-) X be a nonmingled retracting multifunction. Then for each x e X we have x e +(x) and we let G,, be the collection of all distinct sets [+(x) ], x e X. Then G,, is a decomposition of X into disjoint sets and is called the natural decomposition of (generated by) 4. By the discussion in the preceding paragraph, G,, is {isc} if and only if qc is {usc}. Thus we have established the following. THEOREM. The semicontinuity (upper or lower) of a domgposirti.into decomposition into multifunction) multfuntio 5 decomposition ) agrees exactly with that of its natural disjoint sets a eretracting multifunctionf (4) Quasi-Compact Multifunctions.-A nonmingled multifunction f: X Y is quasi-compact provided the image of every open inverse set is open. Since, by (e) of 1, f(x - A) = f(x) - f(a) for inverse sets A cx when f is nonmingled, it is clear that f is quasi-compact if and only if the image of every closed inverse set is closed. Thus "closed" may be substituted for "open" in the definition of quasi-compactness. For mingled multifunctions this would no longer remain valid. Thus we have (4.1) Every open or closed nonmingled multifunction is quasi-compact. Likewise, since compactness is invariant under upper semicontinuous multifunctions, we have (4.2) If X and Y are Hausdorff spaces and X is compact, every upper semicontinuous nonmingled multifunction f: X -- Y is quasi-compact. Not quite so apparent, however, is S

346 MATHEMATICS: G. T. WHYBURN PROC. N..A. S. (4.3) The inverse of every nonmingled upper or lower seinicontinuous multifunction f: X ==) Y is quasi-compact. To see this we note first that f-1: Y==) X also is nonmingled. Let Bc Y be an inverse set for f-1, that is, B = f(a) for some set ACX, and suppose B is closed. Then if f is use, f-'(b) is closed by upper semicontinuity. Also if f is Ise, openness of Y - B gives thatf-1(y - B) is open; and sincef-1(b) = f-1(y) - f1(y - B) = X - f-1(y - B), it follows that f-1(b) is closed. Definition: Iff: X=) Y is a multifunction, a set A cx is a cross section for f if and only if f(a) = Y. (4.4) THEOREM. If a nonmingled multifunction f: X ==) Y is quasi-compact on some cross section A off, thenf is quasi-compact. Proof: Let U be any open inverse set for f in X. Then A. U is an open inverse set for fia: A ==) Y. For if U = f-1(z), ZcY, we have A * U = A *f-1(z) = (fla)-1(z) Hence f(a * U) is open in Y. However, f(a- U) = f(u). For let p e f(u). Then f-1(p) cu because f-1f(u) = U by 1(d); and A f-1(p) is nonempty, since A is a cross section for f. Thus p e f(a * U). (4.5) THEOREM. Every nonmingled retracting multifunction is quasi-compact. For let f: X -) Y be nonmingled and retracting. Set g = fi Y. Then g: Y ==) Y is nonmingled and retracting. Thus since, by (2.5), every inverse set in Y is its own image under g, it follows that fi Y is quasi-compact. Thus our conclusion follows from (4.4) since Y is a cross section for f. (5) Relative Local Connectedness.-A topological space Y will be said to be locally connected relative to a multifunction f: X=) Y provided that components of open image sets in Y are open. Remarks: (i) A space Y which is locally connected relative to any function f: X =) Y is locally connected in the ordinary sense. (ii) A topological space X is locally connected if and only if it is locally connected relative to the identity function on X. (iii) If a space Y is locally connected, then Y is locally connected relative to any multifunction f: X ==) Y whatever. (5. 1) THEOREM. Let f: X =) Y be a nonmingled doubly quasi-compact multifunction which preserves connectedness. Then if f is locally connected relative to f-1, Y is locally connected relative to f. Proof: Being doubly quasi-compact means that both f and f-1 are quasicompact. Our theorem asserts that if components of open inverse sets are open, then components of open image sets are open. To prove this, let U be any open image set in Y and let Q be any component of U. To show that Q is open, let R be the union of all components C of f-1(u) which meet f-1(q). For each such C, f(c) is connected since f preserves connectedness. Thus since by 1(c), f(c)c U and f(c). Q #5A', we have f(c) cq for each such C. Hence R = f-'(q) and f(r) = Q. Now f-(u) is open by the quasi-compactness of f1. Thus R is open because

VOL. 59, 1968 MATHEMATICS: G. T. WHYBURN 347 it is a union of components of f- (U) and X is locally connected relative to f-1. Since R is also an inverse set for f, its image Q is open by the quasi-compactness off. (5.11) COROLLARY. If ff: X ==) Y is nonmingled, quasi-compact, and (upper or lower) semicontinuous and has connected point values, then X locally connected relative to f-1 implies Y locally connected relative to f. (5.12) COROLLARY. If f is nonmingled, retracting, and semicontinuous and has connected point values, the same assertion holds. (5.13) COROLLARY. Iff: X =) Y is a quasi-compact mapping (i.e., single valued and continuous) and if X is locally connected relative to f'1, Y is locally connected. In particular, if X is locally connected so also is Y. Note: The last statement in this corollary is a known result (see Whyburn, G. T., Duke Math. J., 19, 445 (1952)) and it includes as special cases the fact that local connectedness is invariant under all open or closed mappings as well as under all retracting mappings and all quotient mappings. (6) Analysis of Nonmingled Multifunctions.-Let f: X =) Y be any nonmingled multifunction where X and Y are topological spaces. Let 4: X =) X' and At: Y ==) Y' be the natural mappings for the decompositions of X and Y, respectively, into (point) inverse sets and (point) image sets for f. Define h: X'-=) Y' by h(x') = #,f0i-1(x'), x' e X'. Then it is apparent that 4, 46, and h are onto functions and 4 and Vp are continuous and quasi-compact. Likewise, ifwedefineg: X=)Y'ands: Y==)X'by g(x) = 4'f(x), x e X; s(y) =,f-i(y), e Y, then g and s are onto functions. Thus we have the following commutative diagram: f Y Xi4 Y The following additional conclusions are readily verified. (6. 1) (a) h: X' ) Y' is 1 to 1 and is quasi-compact when f is quasi-compact; also h-1 is quasi-compact when f-1 is quasi-compact. (b) h is a homeomorphism when both f and f-1 are quasi-compact. Conversely, if h is a homeomorphism both f and f-1 are quasi-compact. (c) The function g: X ==) Y' is quasi-compact when f is quasi-compact and continuous when f-1 is quasi-compact. (d) The function s: Y =-) X' is quasi-compact when f-i is quasi-compact and continuous when f is quasi-compact. Thus if we define Z to be X' or Y', in turn, and look at the pairs o,s and g,4

348 MA TIEMA TICS: G. T. WHYBURN PROC. N. A. S. we see that in either case f has generated a topological space Z and a pair of functions from X onto Z one of which is continuous and both of which will be quasi-compact and continuous whenever f is doubly quasi-compact. Now it is readily seen, conversely, that if we start with a pair of functions 4: X =) Z, yt: Y =) Z of X and Y, respectively, onto a topological space Z, then f(x) = A-l+(x), x e X, defines a nonmingled multifunction f: X =) Y. Further, f will be doubly quasi-compact if both g and 4, are continuous and quasi-compact; and in this case f will generate a pair of mappings 4/',' equivalent to the given pair Xx, under the procedure outlined above, because h is a homeomorphism. Thus we have (6.2) THEOREM. For any nonmingled doubly quasi-compact multifunction f: X =) Y there exists a topological space Z and a pair of quasi-compact mappings 4: X=- Z and 1: Y=) Z satisfying f = I-10, f-1 = 4-4+. Thus f is closed (open) if and only if 4 is closed (open); equivalently, f is usc (lsc) if and only if ^t1 is closed (open). Conversely, any pair of quasi-compact mappings 4: X =) Z and V: Y =;) Z generate a nonmingled doubly quasi-compact multifunction under the definition f(x) = +-1+(x), x e X, which in turn generates a pair 4/4' equivalent to 4,+. Note: The relation f generated here associates the points (xy), x e X, y e Y such that +(x) = A(y). (*) In other words, y ef(x) if and only if (*) holds. This situation is of especial interest in connection with functions of a complex variable. For suppose z = +(x) and z = P(y) are entire functions without exceptional values where X and Y are complex planes. Then both 0 and 4& are open, and hence quasi-compact mappings onto the plane Z, and the associated multifunction f is defined by the equation (*). For example, consider the equation 2= y3 In this case, for each z E Z the associated multifunction f sends each of the square roots in X of z onto the set of three cube roots of z in Y and, of course, f-1 does just the reverse.