By Irving Adler

This richly specified review surveys the evolution of geometrical principles and the improvement of the options of recent geometry from precedent days to the current. issues comprise projective, Euclidean, and non-Euclidean geometry in addition to the function of geometry in Newtonian physics, calculus, and relativity. Over a hundred workouts with solutions. 1966 edition.

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**Extra info for A New Look at Geometry (Dover Books on Mathematics)**

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We claim that { V i ) : n > 1) is a cover of A . In fact, take any y G A . There is n such that y e B . e. oXj,,6(x )) < j r ( x ) . Since b(x ) € B ( x „ , | r ( x „ ) ) , we have d ( x „ , 6 ( x » < j r ( x „ ) ) . e. y G B ( x , r(x„)) c V . Consequently, {V : n > 1} is a countable open subcover. 9. Theorem Let A be a subset of a metric space X. Then the following statements are equivalent. (a) A is compact. (b) A is complete and precompact. (c) Every open cover has a finite subcover. (d) Every countable open cover has a finite subcover.

Therefore f(A) is separable. 4. Exercise Prove that products of separable sets are separable. 5. Exercise Prove that K " is separable. 6. Theorem Every precompact set A in a metric space X is separable. Complete, Compact and Connected Sets 28 Proof. For each integer n > 1, there is a finite subset J B(x J A C Uc J„ < n>- Let S = I C i n - € A. To show A c S , take any y € A. n of A such that For each n, there is i „ e J that y e B ( x „ , L ) . Then {x„} is a sequence i n B. have x n Then B is a countable subset of Since d{x„,j/) < n such we —• y.

14. Theorem Let A, B be non-empty subsets of a metric space X. If A is compact, then there is a e A such that d(a, B) = d(A, B). Furthermore if A, B are disjoint closed sets, then d(A, B) > 0. Proof. The function d(x, B) is continuous in x on a compact set A. 15. Theorem Let A, B be non-empty compact subsets of a metric space X. Then there are points a e A and be B such that a\a, b) = d(A, B). Proof. Since both A, B are compact, the product set A x B is compact in the product space XxY. Since the continuous function d : A x B —* IR attains its minimum, there is some (a, f>) e A x B such that a\a, b) < dfx, y), V(x, y) e Ax B.