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Extra info for A treatise on the analytical geometry of the point, line, circle, and conical sections
Sample text
I, the intersection B (Ui,)n - - . n B ( U i , ) is complete. 20. Theorem. For every separable metrizable space X, icd X = C-Odim X. Proof. The proof will smoothly follow the steps that were taken in Section 6. 10. 11 will yield the inequality C-Odim X 5 C-def X . The proof of the theorem is completed by an application of the main theorem. 8. THE COVERING DIMENSION dim 41 The results of this section have been generalized in two directions. First, they hold for more general spaces X ; second, they hold for classes of spaces P more general than the class C of complete spaces.
7. Example. e we let 2 = I” \ En-’ with n 2 2. We have already computed d e f Z = n - 1. We shall prove Cmp 2 = n - 1. 6, it will suffice to show that Cmp 2 2 n - 1 holds. To this end, we shall introduce in the next paragraph another invariant that is related to compactness. We write CompX = -1 if and only if X is compact. And for a natural number n we write C o m p X 5 n if for any n 1 pairs (Fo,Go), . . ,(J‘,, G,) of disjoint compact subsets of X there are partitions Si between Fi and Gi in X , i = 0,.
The easy proof will be omitted. The following proposition has a straightforward proof. F. For every closed subspace Y of a separable rnetrizable space X , d e f y 5 defX. 10. Examples. A common feature of many of the examples in dimension theory is that the examples themselves are easy to describe but the values are hard to compute. The same is true for some of the examples given here. a. Define X to be { (z,y) E R 2 : -1 5 x < 1, -1 y 5 l}. The set X is a closed square with a closed edge removed.