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I, is a sequence of natural numbers less than or equal to k. Indeed, if Y has an open shrinking V' which does not satisfy (I), one replaces Y by V' and one continues this procedure until an open 56 Dimension theory of separable metric spaces [Ch. 1, 56 cover with the required property is obtained; as the number of intersections in (1) is finite, the process will come to a n end after finitely many steps. Since o r d Y > n, rearranging if need be the members of “Y-, we have n+2 We shall show that the (n+2)-element open cover {U,>S=+:of the space X , where U, = V, for i < n+ 1 and Un+, = k U i=n+2 V,, has the required property.

Definitions of the large inductive dimension and the covering dimension. Metric dimension The first separation theorem, established in the preceding section, suggests a modification in the definition of the small inductive dimension consisting in replacing the point x by a closed set A . In this way we are led to the potion of the large inductive dimension Ind, defined for all normal spaces. Both dimensions coincide in the realm of separable metric spaces. They diverge, however, in the class of metric spaces and also in the class of compact spaces; let us make clear at once that examples in point are very difficult and will not be discussed in this book.

Let be an arbitrary transformation of T onto 9 ; define a mapping f of T to Z by letting f ( t ) be a point z E Z such that ( t , z) E ( { t }x Z)\p(t) if such Faints exist, and an arbitrary Foint z E Z otherwise. , the graph of the mapping f. 2), for every natural number n define a separable metric space X such that ind X = n and X can be mapped by a continuous and one-to-one mapping onto a zero-dimensional space, Observe that X is totally disconnected and deduce that there exists a totally disconnected separable metric space which cannot be embedded in a Euclidean space.

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