VOL. 53, 1965 MATHEMATICS: L. F. McAULEY 255 9 Groen, P., "Two fundamental theorems on gravity waves in inhomogeneous incompressible fluids," Physica, 14, 294-300 (1948); Groen, P., and B. Heyna, "On short period internal gravity waves," Physica, 24, 383-389 (1958). 10 Eckart, C., Hydrodynamics of Oceans and Atmospheres (New York: Pergamon Press, 1960). See chap. 8. 11 Leighton, R., R. W. Noyes, and G. W. Simon, "Velocity fields in the solar atmosphere," Astro- phys. J., 135, 474 (1962). LIFTING DISKS AND CERTAIN LIGHT OPEN MAPPINGS* BY Louis F. MCAULEY RUTGERS, THE STATE UNIVERSITY, NEW BRUNSWICK, NEW JERSEY Communicated by G. T. Whyburn, December 22, 1964 Introduction.-The concept of lifting mappings is of fundamental importance in the various fiber space and covering space theories. This concept is expressed in a number of ways, but usually as some type of covering homotopy property.' These properties give characterizations of very useful classes of spaces and map- pings related to the following problem. General lifting problem: Suppose that f is a mapping of a space X onto a space Y. Furthermore, suppose that g is a mapping (more generally, g is replaced by a class of mappings, e.g., a homotopy) of a space Z into Y. Does there exist a mapping h of Z into X such that g = fh? For what spaces Z and for what map- pings g: Z -- Y do there exist mappings h: Z -> Y such that g = fh? Under what conditions on X, Y, and f do there exist such mappings h? Here, we shall say that when such an h exists, then h is a lifting of g to X. A special lifting problem: Suppose that Z is an n-cell and that g is a homeo- morphism of Z into Y. Under what conditions on X and f does there exist h: Z X so that g = fh? In case that g is a homeomorphism, we shall say that h is a lifting of g(Z). For example, if Z is an n-cell and g is a homeomorphism, we shall speak of h as a lifting of either g or g(Z). Answers have been provided for special cases of these two problems. Success is obtained by restricting the mappings f:X -- Y to covering maps (X being a covering space of Y') and by requiring the space Y to satisfy conditions such as semilocal simple connectedness.2 A covering map is a local homeomorphism which is a very special kind of light open mapping. A mapping (continuous) f:X -- Y is light iff f-lf(x) is totally disconnected for each x in X and f is open iff for each open set U in X, f(U) is open relative to f(X). At the present time, most definitions of fiber spaces require some kind of covering homotopy property.' Thus, some answers to these problems are built into the structure of the theory. Furthermore, fiber spaces with totally disconnected fibers have fiber mappings which are light and open. Definitions appear in the literatures of generalized covering spaces whose covering mappings are quite special cases of light open mappings. It therefore seems natural to develop some lifting properties for light open Downloaded by guest on September 30, 2021 256 MATHEMATICS: L. F. McAULEY PROC. N. A. S. mappings in an attempt to provide answers to more general lifting problems-a step closer to the two problems stated above. In the case that each of X and Y is a Peano continuum (locally connected com- pact metric continuum), G. T. Whyburn has shown in reference 3 that if f is a light open mapping of X onto Y and h is a homeomorphism of I into Y, then h can be lifted to X. That is, 1-cells may be lifted. In fact, if A is a simple arc (topological 1-cell) in Y, then f-'(A) is the union of a collection G of simple arcs such that ft g is a homeomorphism of g onto A for each g in G. This is remarkable since light open mappings are so much more general than either fiber mappings with 0-dimensional fibers or covering mappings. There is a characterization of open mappings due to Floyd4 which is as follows: Suppose that each of X and Y is a locally connected compact metric continuum and that f: X -- Y is a light mapping of X onto Y. Then a necessary and sufficient condi- tion that f be open is that for each map g:I -> Y where I = [0,1] and for each x in f-lg(O), there exists a map h:I -- X such that g = fh and h(O) = x. Thus, Floyd's characterization says that a light mapping f is open iff all maps g:1 -* Y can be lifted to X (with X and Y as above). In this paper, we shall give some conditions under which a disk (topological 2-cell) in Y can be lifted to X whenever f is a light open mapping of X onto Y. Unless stated otherwise, all mappings are continuous and all spaces are metric. One should observe that there is an at most two-to-one light open mapping of the projective plane X onto a disk Y such that if J = f-1(Bd Y), then fj J is a homeomorphism of J onto Bd Y. However, the disk Y cannot be lifted to X. A Condition Ensuring the Lifting of Disks.-Suppose thatfis a light open mapping of a space X onto a space Y and that D is a disk in Y. We say that (A,B) is a proper pair if and only if B is an arc spanning D (B c D and x E B n Bd D if and only if x is an end point of B) and ft A is a homeomorphism of A onto B. We say that f has Property P relative to D if and only if there is a simple closed curve J in X such that ft J is a homeomorphism of J onto Bd D and f(X- J) n f(J) = 0 and for each E > 0 and proper pair (A,B) there is a a > 0 such that if B' is an arc spanning D and B' c Na(B), then there is a proper pair (A',B') with A' c N,(A). Although Property P is somewhat more complicated than we desire, it is strong enough to yield the following theorem. THEOREM 1. Suppose that f is a light open mapping of X onto Y where each of X and Y is a Peano continuum. Furthermore, suppose that f has Property P relative to some disk D in Y. Then D may be lifted to X. Since no loss in generality results, we assume that Y = D and that D is the unit disk in the plane, i.e., D = (Xy) X2 + y2 < 1}. Hereafter the term "proper pair" will refer only to those proper pairs (A,B) such that B is a straight line interval. We use H(A,A') for the Hausdorff distance between subsets A and A' of a space, i.e., H(A,A') = inf {e > 0tA c N,(A') and A' c NE(A)}. We precede the proof of the theorem by four lemmas. In each we assume the hypothesis of the theorem. Lemmas 2-4 are strengthenings of Property P. LEMMA 1. If I (A,,B)} is a sequence of proper pairs {IB}B B, and {AX}A A, then (A,B) is a proper pair. Proof: Since A is compact, it suffices to show that ft A is 1-1. If this is not the Downloaded by guest on September 30, 2021 VOL. 53, 1965 MATHEMATICS: L. F. McAULEY 257 case, then there are sequences {pi} -* p and {qi} -. q with q,,pi E Ai, p $ q, and f(p) = f(q) = y e B. Now {f(p1)} -- y and {f(qi)} -- y. Let Ai' be the subarc piqi of Ai and Bi' be the subinterval f(pj)f(qi) of Be. Some subsequence { Ai,'} of { A,'} converges to C a subcontinuum of A which contains p and q. Furthermore, since {f(Aki')}-] y, we have f(C) = y. This contradicts the fact that f is light and completes the proof. LEMMA 2. If (A,B) is a proper pair, A pq, and e > 0, then there is an e' > 0 such that if (A',B') is a proper pair, A' = p'q', A' c Nf,(A), d(p,p') < e', and d(q,q') < e', then H(AA') < e. Proof: Suppose that the lemma is not true. Then there is a sequence (An',Bn')} of proper pairs such that (1) An' c Nl1n(A) and A ¢ N,(An') hold for each n, and (2) if An' = pnq", then d(pn',P) < 1/n and d(qn',q) < 1/n. Consider any subsequence { (AknBt) } such that { Ajtj - C and { Bknj -- E. Clearly C c A and I p,q} c C. Since C is connected, we see that E = B and, by Lemma 1, that C = A. From the compactness of X we conclude that { An A and thus reach the contradiction that A c NE(An') for sufficiently large n. LEMMA 3. For each e > 0 and proper pair (A,B) there is a 6 > 0 such that if B' is a straight line interval spanning D and H(B,B') < 6, then there is a proper pair (A',B') with H(A',A) < e.