I 0-regularly. Then

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* This investigation is supported by the National Science Foundation G. 9654. 1 Brauer, R., and M. Suzuki, "On Finite Groups of Even Order Whose 2-Sylow Group is a Quaternion Group," these PROCEEDINGS, 45, 1757-1759 (1959). 2 Feit, W., "On Groups Which Contain Frobenius Groups as Subgroups," Proc. Symp. Pure Math., 1, 22-28 (1959). 3 Feit, W., M. Hall, and J. Thompson, "Finite Groups in Which the Centralizer of Any Nonidentity Element Is Nilpotent," (to appear). 4 Suzuki, M., "On Characterizations of Linear Groups, I," Trans. Amer. Math. Soc., 92, 191204 (1959). 6 Suzuki, M., "On Characterizations of Linear Groups, II," Trans. Amer. Math. Soc., 92, 205219 (1959). 6 Suzuki, M., "A New Type of Simple Groups of Finite Order," these PROCEEDINGS, 46, 868870 (1960). I Thompson, J., Thesis, University of Chicago (1959). 8 Zassenhaus, H., "Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen," Abh. aus dem Math. Seminar der Hamb. Univ., 11, 17-40 (1936).

CONVERGENCE IN NORM* BY G. T. WHYBURN UNIVERSITY OF VIRGINIA

Communicated October 12, 1960

1. Introduction.-By the norm N(f) of a mapping f(X) = Y is meant the least upper bound of 6[f-(y) J for y e Y. Here X and Y are metric spaces and b(K) is the diameter of K for any set K. We shall be concerned with sequences of mappings (f,,) which not only converge in the usual way to a mapping f, but also converge in norm in the sense that N(fn) 0 as n - a. A sequence of sets Yn is said to converge 0-regularly to a set Y [written Yn-*i Y 0-regularly] provided that Yn -- Y in the sense lim sup Yn = lim inf Yn = Y and also for any E > 0 a 6 > 0 and an integer N exist such that if n > N, any two points x, y e Yn with p(x, y) < 6 lie together in a connected subset of Yn of diameter < e. All spaces are assumed to be metric; and for any open set U in a space, Fr(U) denotes the boundary U - U of U. Also, 1D denotes the empty set. 2. THEOREM. Let the sequence of closed mappings fn(X) = Yn C Z converge almost uniformly to a mapping f(X) = Y, where X is locally compact and Yn -I Y 0-regularly. Then if N(fn) -* 0 with 1/n, f is quasi open, and if f-1(y) has a nonempty compact component for each y e Y, f is compact and monotone. This will be a direct result of the following. LEMMA. For any compact component K of f-'(y), for y e Y, and any open sot V in X containing K, there exists a compact set H with K c int H c H c V, an integer N and an open set R in Z about y such that

fn(X

-

H) R

=

F, for all n > N.

To obtain the theorem from this lemma we note first that f(X - H) -R must also be empty by convergence of fn to f, so that

f(V)

n

f(H) v R*Y

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and f is quasi-open. Further, since V is an arbitrary neighborhood of K and we have f'l(y) c V, this gives f-1(y) = K. Thus f is monotone if each f-'(y) has a compact component; and since any quasi-open monotone mapping is compact, f must also be compact under these conditions. Now to prove the lemma, let U1 be an open set in X containing K with U1 compact, U1 c V and Fr(Ul) -.f(y) = (D. Then take an open set U2 v Ui so that H = U2 is compact, H c V and (H - U1) .-f(y) = (D. Since N(fn) 0 and p(U1, X -U2) > 0, there exists an integer N1 such that (i) fn(U1) fn(X - U2) = 4) for all n > N1.

Now let 6e = p [y, f(H - U1) ]. Then by uniform convergence of fn to f on the compact set H - U1, there exists an integer N2 such that (ii) fn(H - U1) * V3E(y) = (D for all n > N2. By regular convergence there exists a 6 > 0, 6 < E, and an integer N3 such that any two points p, q, e Y. with p(p,q) < 6 and in n > N3 lie together in a connected subset C of Yn with 6(C) < e. Now let N4 be chosen so that fn(K) Va/2(y) $ (D for n > N4. Finally, we define N = N1 + N2 + N3 + N4 and R = V512(Y). We must show that fn(X - H) R = 4) for n > N. Suppose, on the contrary, that this set contains a point q for some fixed n > N. Then, since n > N4, fn(K) * R contains a point p. However, p(pq) < 6, and since n > N3, there exists a connected subset C of Yn containing p + q and with b(C) < e. Also, C c V3E(y), because p(p,y) < e; and thus C fn(H - U1) = 4) by (ii). This gives C c fn(U1) + fn(X - U2), with C intersecting the sets on the right in at least p and q respectively. However each of these sets is closed by closedness of fn, and their intersection is empty by (i). This contradicts the connectedness of C. Thus the lemma is established. COROLLARY. Let the sequence of mappings fn(X) = Y converge uniformly to the mapping f(X) = Y where X is compact and Y is locally connected. Then if N(fn) 0 with 1/n, f is monotone. For, in this case, Y is a locally connected continuum and Yn Y, so that the 0regularity condition is automatically satisfied. 3. THEOREM. Let the sequence of mappings fn: X -- Z converge (pointwise) to the mapping f(X) = Y, where X is a locally connected generalized continuum and suppose N(fn) -O 0 with 1/n. Then if X is cyclic, any two points of Y are conjugate in Z. NOTE. A connected set is cyclic if it has no cut point; and two points a and b in a space Z are conjugate provided they cannot be separated in Z by any single point of Z. To prove the theorem, we suppose, on the contrary, that some point y in Z separates two points of Y in Z so that we have a separation Z - y = Z1 + Z2 4) where Y - Z1 Y Z2. Let A = Y - Z1, B = Y Z9. Our conclusion follows at once from the two assertions: (i) If L = lim sup fn-1(y), then L $ 4). (ii) Any p e L is a cut point of X. -

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Now to verify (i), we have only to take a E A, b E B, a' ef-'(a), b' ef-(b) and let a'b' be any simple arc in X from a' to b'. Then, since fn(a') -i a and fn(b') -b, we have fn(a') e Z1, f.(b') E Z2 for all n sufficiently large. Thus fn(a'b') D y and a'b' -fn-1(y') $6 4! for all such n, so that L a'b' (D. To show (ii), let p e L and let a e A and b e B be chosen so that a $ f(p) $ b. Then if a' e f-1(a), b' E f-1(b), every arc a'b' in Z intersects almost all the sets fn -(y) as just shown in the preceding paragraph. However, since a[fj-'(y)] - 0 with 1/n. this gives p e a'b'. Thus, p lies on every arc in Z from a' to b', so that p separates a' and b' in Z. COROLLARY. Let the sequence of mappings fn(X) = Y converge (pointwise) to the -mapping f(X) = Y, where X and Y are locally connected generalized continua, and suppose N(fn) -O 0 with 1/n. Then, if X is cyclic, so also is Y. 4. Application to mappings on the 2-sphere.-It now follows that if X and Y are locally connected continua, with X cyclic, and fn(X) = Y is a sequence of mappings converging uniformly to f(X) = Y and with N(fn) -- 0, then f is monotone and Y is cyclic. In this form the result is of especial interest in case the domain space X is the 2-sphere. For it then follows that Y is cyclic and also is the monotone image of the 2-sphere so that it must itself be a (topological) 2-sphere. Hence Y is homeomorphic with X in this case. It should be noted, however, that the limit mapping f(X) = Y need not be a homeomorphism even in this case. For we can easily define a sequence of homeomorphisms of a sphere onto itself which converges uniformly to a monotone mapping which is not a homeomorphism. Indeed, by a theorem of Youngs,' every monotone mapping of the 2-sphere onto itself can be uniformly approximated by homeomorphisms. By our results above, no other type of mapping on the 2sphere can be uniformly approximated by onto mappings of any sort with norms converging to 0. Thus if fn(X) = Y converges uniformly to f(X) = Y and N(f.) -i 0, where X is a 2-sphere, for any e > 0 there exists a homeomorphism h(X) = Y with p(fnh) < e for almost all n. 5. Uniform openness.-A sequence of mappings f.(X) = Y is uniformly approximately open2 provided that for any E > 0 a 8 > 0 and an integer N exist such that for any x e X and any n> N,fj[VE(x) ] D V8[fn(x)]. Clearly, this is equivalent to the condition that for any e > 0 a 8 > 0 and integer N exist such that for x, y E Y, p(x,y) < 8 and n > N imply that fn-j(x) c VE[fn-'(y)] and (automatically) fn-j(y) c VE[ffn-(x)]. Now if X and Y are compact and the sequence fn(X) = Y converges uniformly to the mapping f(X) = Y, uniform openness implies at once that for any E > 0 we have fl(y) c VE[fn-I(y)] and fn-I(y) c VE[f-I(y)] for all y e Y and n sufficiently large. Thus we have lim f'-I(y) = f-l(y) for -

nf- co

each y e Y. Compare this form with Polak3). Hence we have immediately the THEOREM. Let the sequence fn(X) = Y converge uniformly to f(X) = Y where X and Y are compact and suppose the sequence (fn) is uniformly approximately open. Then if N(fn) 0 with 1/n, f is a homeomorphism. This theorem together with an earlier result of the author2 gives the

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COROLLARY. Let the sequence of homeomorphisms hn(X) = Y converge uniformly to a mapping h(X) = Y. In order that h be a homeomorphism it is necessary and sufficient that the sequence (ha) be uniformly approximately open. * This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract AF 49(638)72 at the University of Virginia. I Youngs, J. W. T., "Homeomorphic approximations to monotone mappings," Duke Math. Jour., 15, 82-94 (1948). 2 Whyburn, G. T., "Sequence approximations to interior mappings," Am. Soc. Polon. Math., 21, 147-152 (1948). 3 Polak, A. I., "On mechanisms of uniform approximations in the domain of continuous mappings of compacta" (Russian), Uspekhi Mat. Nauk (NS), 11, No. 4 (70), 149-154 (1956).

SIGNIFICANCE OF CONTINUED VIRUS PRODUCTION IN TISSUE CULTURES RENDERED NEOPLASTIC BY POLYOMA VIRUS* BY RENATO DULBECCO AND MARGUERITE VOGT CALIFORNIA INSTITUTE OF TECHNOLOGY

Communicated by G. W. Beadle, October 24, 1960

The infection with polyoma (PY) virus of monolayer cultures of either mouse or hamster embryo cells gives rise to a neoplastic transformation of the cultures.' Previous results showed that the transformed hamster cultures did not release any virus, whereas the transformed mouse cultures continued to release PY virus indefinitely; in the latter case, the virus released was invariably a mutant of smallplaque (sp) type.2 We have since observed that in some cases transformed hamster cultures also continue to release virus indefinitely and that the virus released is also of the sp type. These observations raise a number of questions: Why do transformed cultures release virus indefinitely in some cases and not in others? Why is the released virus of these cultures, once transformed, invariably the sp mutant? How is the release of virus related to the transformation? To try to answer these questions, we have studied in detail the evolution of the infected cultures, the properties of the transformed cells, and the quantitative aspects of virus release from transformed mouse embryo cultures. Results.-Growth characteristics of hamster and mouse embryo cultures transformed by PY virus: One of the main in vitro characteristics of transformed mouse or hamster embryo cultures is an increased "over-all growth rate," as compared to that of noninfected cultures. A measure for the "over-all growth rate," i.e. the average rate at which a culture grows when maintained by serial transfer, was obtained as follows. All the cultures were grown as monolayers in contact with the bottom of petri dishes and were transferred without delay as soon as their cells had grown into a confluent sheet. At each transfer, a known fraction of the trypsin-dispersed culture was used as the inoculum. Thus, the increment in cellular mass at every transfer was known, as well as the time required to achieve it. Plotting the logarithm of the product of all increments at successive transfers versus