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A theorem on the Lebesgue dimension

1950; Institute of Mathematics of the Czech Academy of Sciences; Volume: 075; Issue: 2 Linguagem: Inglês

10.21136/cpmf.1950.120779

1802-114X

Miroslav Katětov,

Advanced Banach Space Theory

In a recent paper [1]*) of the present author, some results have been established concerning the relations between the inductive (Menger-Urysohn) dimension of a compact (= bicompact) space P and certain properties of the ring C(P) consisting of all (bounded) continuous realvalued functions on P. In the present note I intend to give a characteri zation of the Lebesgue dimension (in a sense slightly different, for non-normal spaces, from the usual one) in terms of the ring C(P), namely, to show that the Lebesgue dimension of P is equal to the analytic pseudodimension of P, to be defined in the sequel.§i-We first summarize some definitions and results given in [1].-Space always means a Hausdorff topological space, mapping means a continuous transformation, function means a real-valued function.The letter-P denotes a (non-void) completely regular space, R denotes a metric space.' ' '.Let C be a commutative ring (with a unity element) in which there is defined, for any xe C and any real number A, the multiple Xxe C satisfying the usual axioms, and let C be, at the same time, a topological space such that the operations x + y, xy, he are~continuous.Then C will be called a (real commutative) analytic ring (with a unity).We shall say that a subring C x 3 C is algebraically closed (in C) if (1) C x is an ana lytic subring, i.e. contains all Ae where A is real, e is the unity element of C, (2) x e C is contained in C x whenever x n -f-a^^1 -f-... + a n == 0, tti € C x \ if, moreover, C t -= C x (i.e. C' x is a closed set) we shall say that C x is analytically closed (in C). -If'P is a completely regular space, then C(P) denotes the analytic ring consisting of all bounded continuous functions / in P (with the topology defined by the norm |/[ =sup tc p|/(<)|).*) The number in brackets refer to the list at the end of the paper.1 ) This notion is different from E. Hewitt's [4] notion of <& "$et of analytic generators"..2) Thus dim has, in this note, two different meanings: 1. the analytic dimension of an analytic ring, 2) the Lebesgue dimension of a space, to be defined below.