(a) If om = [MI is a directed family in X, 9i = f(m) ], M E An, is a directed family

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1 DIRECTED FAMILIES OF SETS AND CLOSEDNESS OF FUNCTIONS* BY G. T. WHYBURN UNIVERSITY OF VIRGINIA Communicated July 12, Introduction.If X and Y are topological spaces, a (singlevalued) transformation or function f: X Y is compact provided the counterimage f1(k) of every compact set K in Y is compact; and f is closed provided the image f(c) of every closed set C in X is a closed set in Y. It is well known (see refs. 1 and 2) that for the more usual spaces Y, for example, weakly separable metric, a mapping (i.e., a continuous function) of X into Y is compact if and only if it is closed and has compact point inverses. Thus the two properties of a function, (A) compactness and (B) closedness plus compact point inverses, are equivalent for mappings into such range spaces and even less restricted ones, although they are not always equivalent. Property (B) implies (A) for any space Y, but not conversely. In this paper it will be shown first how the property (B) for functions, not necessarily continuous, may be characterized in terms of directedness of families of sets. It turns out that a function has (B) if and only if directedness of a family toward a given set is preserved under the inverse f1 of f. Results are then developed showing the conditions on a range space Y under which (A) and (B) are equivalent for mappings into Y and for arbitrary functions into Y in case both Y and the domain space are Hausdorff. 2. Preliminaries.A nonempty collection of nonempty sets f in a topological space is called a directed family, provided the intersection F1 F2 of any two elements F1 and F2 of 9 contains some elements F3 of W. A directed family 5f is said to cluster at a point p, and p is then a cluster point of 5, provided every open set U containing p intersects each element F of 5. Also, 9: converges to a point p, provided every open set U about p contains some element of 5. A directed family 5' will be called a directed under family of a directed family 5, provided every element F of 5: contains a subset F' which is an element of 5'. Finally, a directed family 5 will be said to be directed toward a set E, provided every directed under family of 5 has a cluster point in E. (Note: No directed family can be directed toward the empty set.) Now letf: X * Y be a singlevalued function, not necessarily continuous. (Note: Continuity is implied only when the term mapping is applied to f.) We record next a number of readily established facts. (a) If om = [MI is a directed family in X, 9i = f(m) ], M E An, is a directed family in Y. (b) If S9 = [N] is a directed family in f(x), M = [fi(n) ], N e 9X, is a directed family in X. For any nonempty set E in X and any directed family Si in f(e), [Ef1(N)], N e 9X, is a directed family in E. (c) If em = [M] is a directed family in X, S9 = [f(m) ], M E A, 9i' is a directed under family of Si and SA' = [fi(n') ], N' E Si', then the collection of sets S" = [M M'] for all M E S and M' E S' is a directed under family both of Sit and of Si'. To verify this, we note first that each such set M" = M M' is nonempty. For f(m) contains some N1' E Si' and f(m') is itself an element of Si'. Thus N1'.f(M') 688

2 VOL. 54, 1965 MATHEMATICS: G. T. WHYBURN 689 contains some N' E 91'. Hence f(m) f(m') D N' so that Mfl(N') * 4) and fl(n')cflf(m') = M'. Thus MOM' * (D. (Note: (D denotes the empty set.) Next, 1" is a directed family. For take Ml = Ml Ml', M2' = M2*M2'. There exist M3 em with M3c M1 M2 and M3'3 e ' with M3'cM1' M2'. This gives M31 = M31M3'c(MlM2).(Ml3M21) = (MlMl')(M2M2') = M1U1M2ff. Finally, Ad is clearly an under family of both 9 and 1', because any ME1 and also any M' c 91' contains the element M13M' of 11'. (d) A directed family 5 in a space X converges to a point p if and only if F is directed toward p. Proof: For if 5 converges to p, every open set U about p contains a member of 9 and thus contains a member of any directed under family 51 of 5, so that 51 actually converges to p. On the other hand, if 5 is directed toward p, it must converge to p. For if not, there exists an open set in X about p which contains no element of W. Denote by 5' the family of sets F' = F (X U) for F e 5. Then the sets F' are nonempty. Also 5' is a directed family and indeed it is an under family of 5, because given F1' = F1(X U) and F2' = F2(X U), there is an F3cFlF2 and this gives F3' = F3.(X U)cFl.F2(X U) = F1(X U)F2(X U). Byconstruction p is not a cluster point of V'. This is a contradiction, and thus 5 converges to p. (e) Given Bc Y. Iffor each directed family Di inf(x) directed toward a point p e B, the inverse family 9 = [f'(n)], N E 91, is directed toward f1(p), then for any directed family if of sets in f(x) directed toward a set B, 8 = [f'(f) ], F E i, is directed toward A = f1(b). Proof: Under this hypothesis any p E B which is a cluster point of an under family of 5 must be in f(x). Thus not only is B.f(X) $ 4), but also 3f is directed toward B f(x). Thus we may assume Bcf(X). Let 91 be a directed under family of 8. Then DI = [(M) ], M e 1, is a directed under family of 5 by (a). Thus 9a has a cluster point y in B and a directed under family a' of 91 converges to y and thus is directed toward y. By hypothesis 1' = [(f1(n') ],N' e 91', is directed toward f'(y). Also by (c), 911 and 911' have a common directed under family 9". Thus 911 has a cluster point p in f(y). Since p is then a cluster point of 911 and p e f1 (y) ca, our conclusion follows. 3. THEOREM. A function f: X Y is closed and has compact point inverses if and only if for each family 3f in f(x) directed toward a set B in Y, the inverse family E = [f'(f) ], F E 5, is directed towardfl(b). Proof: Suppose f is closed and has compact point inverses. Then by (d) and (e) it suffices to show that if 91 is a family of sets in f(x) converging to a point y in B, then 911 = [f(n)], N e 91, is directed toward f'(y). Suppose to the contrary, that for some directed under family 9' of 911, no point of fl(y) is a cluster point of 911'. We show, however, that this leads to the contradiction that the directed under family 91' = [f(m') ], M' e 9', of 91 cannot converge to y. For each x f 1(y) by supposition there is an open set Ux about x and MA1' E 911' with MV' Us = 4). Since f(y) is compact, it is contained in a finite union U = Uf of the sets U7. Let M' be an element of 9' which is contained in the intersection srm, ' and let V be the open set Y f(x U). Then f(m') * V = 4) because M'cX U. Thus since f(m') e 91'. 91' cannot have y as a cluster point.

3 690 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. Now suppose our condition is satisfied but f is not closed. Let E be a closed set in X such that some y e Y f(e) is a limit point of f(e). Let Ot be the directed family of sets f(e) * V for all open sets V in Y containing y. Then 9t is a directed family in f(x) converging to y. Let 1 = [f'(n)], N e OT, and M1Z' = [E.M], M e M. It readily follows that M' is a directed under family of M. But since X E is open and contains fi(y), St' has no cluster point in f'(y). This is a contradiction, and thus f must be closed. Finally, to show each f(y) is compact, we have only to show that every directed family of subsets of f'(y) has a cluster point in f1(y). This is trivial for y e Y f(x). Also for y e f(x), {yi is a directed family in f(x) directed toward y. By hypothesis, {f'(y)} must be directed toward f'(y). This means that every directed family of sets inf1(y) has a cluster point inf'(y), so thatf1(y) is compact. COROLLARY 1. A function f: X Y is closed and has compact point inverses if and only if each directed family in f(x) converging to y e Y has inverse family directed toward f'(y). COROLLARY 2. Iff: X Y is closed and has compact point inverses, the inverse of any compact set in Y is compact. Proof: For if K is any compact set in Y and M is a directed family in f'(k), 9 = [f(m) ], M E a, is a directed family in K and in f(x) and is directed toward K. Thus [f1(n) ], N e N, is directed toward f1(k) so that its directed under family Mnz has a cluster point in fi(k). 4. Compactly Closed Sets and Compact Mappings.Definition: A subset E of a topological space X is said to be compactly closed, provided its intersection with every compact set is compact. Remarks: This notion, but not the term, has been attributed to Hurewicz by Gale3 who calls a space a kspace provided every compactly closed set is closed. Also see Kelley4 where an almost identical kspace is devel oped. We note further: (1) every closed set is compactly closed; (2) in a Hausdorff space, a set is compactly closed if and only if its intersection with every compact set is closed; (3) in a weakly separable or locally compact Hausdorff space, every compactly closed set is closed; (4) in a Hausdorff space every compact set is compactly closed. However, in a nonhausdorff space this does not necessarily hold. The following is an example of a perfectly separable, countable, compact space which is the union of two compact sets whose intersection is not compact, and therefore neither is compactly closed. Example: Let X consist of a sequence {Is, }I =i plus two distinct points a and b not in {si}. An open set about a (or b) is a plus all but finitely many si. On XO co X a b assign the discrete topology. Then S, = a + E si and S2 = b + fi si are compact sets, but S1 * S2 = si is not compact. i= 1 THEOREM. Given a topological space Y, in order for all compact mappings f: X Y of a topological space X into Y to be closed it is necessary and sufficient that every compactly closed set in Y be closed. Further, if X and Y are Hausdorff spaces and Y satisfies this condition, all compact functions f: X k Y are closed. Proof: We prove two propositions from which the theorem follows.

4 \OL. 54, 1965 MATHEMATICS: G. T. WHYBURN 691 (1) For any compact function f: X Y, the image of every closed set is compactly closed provided either (a) f is continuous or (b) X and Y are Hausdorff spaces. (2) If Y is a (Hausdorff) space which contains a compactly closed set which is not closed, there exists a (Hausdorff) space X and a 1 1 mapping f: X * Y of X into Y which is compact but not closed. Proof of (1): Let A be any closed set in X. We have to show that if K is any compact set in Y, then K.f(A) is compact. Now the set H = A.f1(K) is a closed subset of f1(k) and f'(k) is compact. Hence H is compact. Since f(h) = K f(a), this latter set is compact in case (a), where f is continuous. In case X and Y are Hausdorff spaces, we suppose the set E = f(h) = K.f(A) is not compact, and thus not closed, and let p e E. The directed family of sets 9Z = EK U for all open sets U in Y containing p clearly converges to p. The inverse family 9R = [A f'(n) ],N e X, is contained in the compact set H and thus has a cluster point q in H. Now q' = f(q) * p since p is not in f(a). Thus there exists an open set V about p whose closure V7 does not contain q'. Then f'(k V) = B is a compact (and therefore closed) set in X not containing q. Accordingly, X B is open and contains q but fails to intersect the element [A *fl (E V) ] of SR. Thus our supposition that E is not compact leads to a contradiction. Proof of (2): Let H be a subset of Y which is compactly closed but not closed in Y. Let p EF H. Define X to consist of points in the set H' + p' where H' = H and p' = p; and let a basis for the topology' consist of (p') together with all subsets U' of H' such that U is open relative to H in Y, i.e., there exists an open set V in Y such that V H = U. Let f: X Y be defined byf(x') = x for x'e X. Thenf is continuous. For let V be any open set in Y. Then V. H = U is open relative to H and thus U' is open in X. Also f1(v) consists of U' + p' or of U' according as V does or does not contain p. In either case f1(v) is the union of open sets and thus is open. Also f is compact. Let K be any compact set in Y. Then K H is compact. To show f'(k) is compact, it suffices to show that f1(k H) is compact, because f'(k) consists of fi(k.h) + p' orf'(kh) according as K does or does not contain p. Take an open covering [U'] of f'(k H) by basic open sets U'. Then [U] is an open covering of K. H, thus reducible to a finite cover, ENg. The corresponding union U1' + U2' U,' is a finite subcover of [U'] covering (K AH). Thus f is compact. However f is not closed because X p' is closed but f(x' p') = H and H is not closed. It will be noted that if Y is a Hausdorff space, so also is X. Hence the statement of (2) is valid with or without the parentheses. Remarks: A space X has Property a (compare Halfar5 and Whyburn6), provided that whenever a point p is a limit point of a set H, there exists a compact set KCX such that p is a limit point of H. K. We readily verify the ASSERTION. If X is a Hausdorff space that has Property a, then every compactly closed set is closed. For suppose X contains a compactly closed set H which is not closed. Let p e H H. There exists a compact set K in X such that p is a limit point of K H. However, K H is compact but not closed, which is impossible in a Hausdorff space. Example: The Hausdorff separation property is essential in the above assertion. Let X = I a,, a2,... I be a sequence of distinct points. A basis for the open sets is

5 692 PHYSICS: R. SERBER PROC. N. A. S. the collection { aj + E an }. Then every set is compact, but not every set is closed n =,m (e.g.! X a is not closed). Thus X has Property a but contains a compactly closed set that is not closed. * This research was supported by a grant from the National Science Foundation. 1 Whyburn, G. T., "Open mappings on locally compact spaces," Am. Math. Soc. Memoirs, no. 1 (1950). Also "Open and closed mappings," Duke Math. J., 17, 6974 (1950). 2 Halfar, E., "Compact mappings," Proc. Am. Math. Soc., 8, (1957). 3 Gale, D., "Compact sets of functions and function rings," Proc. Am. Math. Soc., 1, (1950). 4 Kelley, J. L., General Topology (New York: D. Van Nostrand, 1955), p. 230 ff. 6 Halfar, E., "Conditions implying continuity of functions," Proc. Am. Math. Soc., 11, (1960). 6 Whyburn, G. T., "Mappings on inverse sets," Duke Math. J., 23, (1956). SHADOW SCATTERING AT LARGE ANGLES* BY R. SERBER COLUMBIA UNIVERSITY, AND BROOKHAVEN NATIONAL LABORATORY, UPTON, NEW YORK Communicated July 28, 1965 As the energy of an elastically scattered particle is increased until its wavelength, 1/k, becomes small compared to the dimensions of the scatterer, it would be expected that the phase shift 5I should depend on the ratio p = (1 + '/2)/k, that is, that as k is increased, the scattering takes place at fixed impact parameter p, rather than at fixed angular momentum 1. This result is just what would be given by solution of a wave equation by the WKB approximation, appropriate for small wavelength (it is, of course, the WKB approximation which dictates the proportionality of p to 1 + l/2, rather than to 1 or [1(1 + 1) ]V/'). In these circumstances many terms contribute to the sum for the scattering amplitude, f _ CD ik 2k2 g (21 + 1) a(l)p(z), (1) (where a(l) = 1 e2iw and z = cos 0), and a frequently used approximation is to replace (1) by f co ik = J A(p)Jo(qp)pdp, (2) with q the momentum transfer, q = 2k sin 0/2, and A(p) = A([l + 1/2]/k) = a(l). (3) Equation (2) is obtained from (1) by replacing the sum by an integral, and approximating P1 (cos 0) by Jo([ ] sin 0/2). While these approximations are very good for small angles, at large angles the equality between the Legendre and Bessel functions is in error by terms of order sin2 0/2, and moreover the scattering given by (2) becomes so small that it may be comparable to the error in replacing the sum by an integral. In this note we shall investigate the relationship between the results