Spaces of Uncountably Many Dimensions*

Riemann in his Habilitations Schrift of 1854 suggested the notion of ^-dimensional space (where n is a natural number) as an extension of the notion of three-dimensional euclidean space. Hubert extended the notion still further by defining a space of a countably infinite number of dimensions. Fréchetf in 1908 defined two other spaces of countably many dimensions, which he called D„ and J3W. TychonoffJ in 1930 defined a series of spaces of an unlimited number of dimensions and established several of their properties. The present paper undertakes, by generalizing the notions of spaces A> and EU1 to define spaces D and E a , respectively, for each cardinal number \Aa representing the number of dimensions. I t is shown that every metric space is homeomorphic with a subset of some space D. Certain properties of Tychonoff's spaces, here called spaces T, are also presented. 1. Spaces D. For each initial ordinal number œa the set of all points of space D is the set of all type œa sequences [x;]/°« of real numbers Xi, such that 0Ney then |:y—#*,,-| <e for every value of j<co a . The point P is said to be the sequential limit point of [Pi]/. A point P is said to be a limit point of a point set M, provided there exists a sequence of distinct points of M which converges to P. Thus space D° is equivalent to space D„ of Fréchet. For each two distinct points A = [xi]f" and B = [*••]»"« of D such that for each i, x^Zi, let D"A,B)J also referred to as segment AB, denote the set of all points P=[yi]f such that Xi<yi<Zi or Zi<yi<Xi. If there exist constants h and k, (h<k), such that for each i, Xi = h and yi = k, the notation D"htk) is used. It is evident that D"htk) is homeomorphic with D a for every value of a, h, and k\ but there exist points A and B such that D"A,B) is not homeomorphic with D.