1617-257/TUT-R-8

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On 11/3/16, we discussed some questions from the exam:

Problem 1. Let E be an infinite subset of a compact metric space X. Show that E has a limit point.

Proof. If E has no limit points, then E is a closed subset of a compact space and is therefore compact in itself. Since each point of E is isolated, we may find for each point e \in E a neighborhood U_e such that  E \cap U_e = \{ e \}. The collection \{ U_e\}_{e \in E} is an open cover of E which clearly has no finite subcover.

Problem 2. Let f: B_1(0) \to \mathbb{R}^2 be a function which is "jelly-rigid": for all x,y \in B_1(0): |f(x) - f(y) - (x - y)| \leq 0.1 |x - y|. Prove that f maps onto B_{0.4}(0).

Proof. Since f is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on B:= \overline{B_1(0)}. A simple estimate shows if f(x) \in B_{0.4}(0), then x \in B_1(0). Suppose now that there is some point z \in B_{0.4}(0) which is not in the image of f. Let x_0 \in B_1(0) be a closest element to z in the image of f. Consider now the point x_1 := x_0 + \delta (z - f(x_0)) (jelly-rigidity says that the function is almost like the identity, so moving closer to the point z from f(x_0) in the codomain side can be obtained by moving in that same direction on the domain side first) where \delta > 0 is chosen to be small enough so that x_1 \in B_1(0) and 0< \delta < 1. Then f(x_1) is closer to z than is f(x_0).