On $k$-to-1 transformations

W. H. Gottschalk
1947 Bulletin of the American Mathematical Society  
The following results are extensions of certain of the theorems of O. G. Harrold {Exactly (k, 1) transformations on connected linear graphs, Amer. J. Math. vol. 62 (1940) pp. 823-834). Let X and F be compact Hausdorff spaces and let ƒ be a continuous transformation of X onto F. Let k be a positive integer and let pE denote the cardinal of the set E. We say that ƒ is at most k-to-i (or exactly k-to-1) in case y£F implies jjf~l(y)^k (or fif^iy) =&). Let o{x) denote the order of the point x. That
more » ... s to say, o(x) is the smallest integer m such that fx bdy U=m for an arbitrarily small open neighborhood U of x, if such exists; otherwise o(x) is oo.
doi:10.1090/s0002-9904-1947-08775-9 fatcat:b7pbkpefyzetdb3a4pmkw5fcqa