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INIDEPENDENT UNIVERSITY OF MOSCOW

2. TOPOLOGICAL AND METRIC SPACES

TOPOLOGY-1, SPRING 2012 Г.

of a Hausdor space is closed; (c) the image of a compact space under a continuous map is compact. Problem 2. Let A Rn be a closed subset, let C Rn be a compact subset. Prove that there exists a point c0 C such that d(A; C ) = d(A; c0 ). Further, prove that if the set A is also compact, then there exists a point a0 A such that d(A; C ) = d(a0 ; c0 ). Show that if A and C are closed but not compact then both statements may be false. Hint. A subset K R is compact if and only if it is bounded and closed. Now combine 1(c) with Problem 1.5. = {(x1 ; x2 ; : : : ; xn ) Rn | x1 ; : : : ; xn 0; x1 + · · · + xn = 1} is compact and the function f : R given by the formula f (x1 ; : : : ; xn ) = x1 : : : xn is continuous. (The topology of is inherited from Rn .) (b) Prove that the function f achieves its maximum at the point (1=n; 1=n; : : : ; 1=n). Prove that for all z1 ; : : : ; zn 0 the inequality n z1 : : : zn (z1 + · · · + zn )=n holds. Problem 4. Prove that (a) the image of a connected space under a continuous map is connected; (b) the same, for path connectedness; (c) an open subset in Rn is connected if and only if it is path connected. Problem 5. (a) Prove that the topological space SO(3) is path connected. (b) Prove that the topological space GL(n; R) is not path connected. (c) Prove that the topological space GL(n; R) is a disjoint union of two path connected components. Problem 6. A set A R2 is a union B C where B is a unit circle centered at the origin, and C is given in polar coordinates (r; ') by the equation r = '=(1 + '), 0 ' < . Prove that A is connected but not path connected. Problem 7. Prove that (a) d(x; y) = max1in |xi - yi |, (b) d(x; y) = 1in |xi - yi |, where x = (x1 ; : : : ; xn ) and y = (y1 ; : : : ; yn ), is a metric in Rn . Problem 8. For a rational number x Q, x = 0, denote by x 2 the number 2-k where k is an integer (positive, negative or zero) such that x = 2k m where m; n Z are odd. Take also 0 2 = 0 by de nition. (a) Prove that n d(x; y) = x - y 2 is a metric in Q; denote the metric space obtained Q2 . (b) Is Q2 compact? (c) Is the subspace Z Q2 compact? (d) What subsets of Q2 are connected? Let l1 ; l2 ; : : : ; ln-1 ; d be a plane hinge mechanism that consists of n rods, one of which is xed and the other rods (together with the xed rod) form a closed polygonal line, with xed rod of length d and moving rods of lengths l1 , l2 , . . . , ln-1 numbered successively. Problem 9. Find the con guration spaces of the following quadrangles: (a) 1; 1; 1; 2:9 ; (b) 1; 1; 1; 1 ; (c) 2; 3; 2; 3 .

Problem 1. Prove that (a) a closed subspace of a compact topological space is compact; (b) a compact subspace

Problem 3. (a) Prove that the topological space

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