By Courant R., John F.
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Extra resources for Introduction to calculus and analysis. Vol.2
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Local Compactness 41 EXERCISES Show that the unit circle S 1 in � 2 and the unit interval [0, 1] both are (Hausdorff) compactifications of �. Hint: Use the fact that � is homeomorphic to the open interval ]0, 1 [ and (therefore also) homeomorphic to S 1 \ {(I, O) }. 2. Show for every n in 1\1 that � has the closed unit ball B n = n n {x E � l xf + . . + x; ::;; I } in � as a compactification. 3. Show for every n in 1\1 that � has the unit sphere s = n+ n+ . . 1 1 {x E � 1 xf + + X;+ 1 = I } in � as a compactification.
Product topology. Final topology. Quotient topology. Exercises. 1. Let (X, 't") and ( Y,a) be topological spaces. A function f: X -+ Y is said to be continuous if f- 1 (A) E 't" for every A in a . It is said to be continuous at a point x in X if f- 1 (A) E @(x) for every A in @(f(x» . 2. Proposition. A function is continuous iff it is continuous at every point. PROOF. If f: X -+ Y is continuous and A E @(f(x» for some x in X, choose B e A in @(f(x» n a. Then f- 1 (B) E 't" and f- 1 (B) c f- 1 (A), whence f- 1 (A) E @(x).
18. Show that the unit circle in 1R 2 is not homeomorphic to any subset of IR. 4. 17. 17. 19. ) Two continuous maps f: X -+ Y and g : X -+ Y be tween topological spaces X and Y are homotopic if there is a contin uous function F: [0, 1] x X -+ Y (where [0, 1] x X has the product topology), such that F(O, x) = f(x) and F(l, x) = g (x) for every x in X. Intuitively speaking, the homotopy F represents a continuous de formation of f into g. Show that any continuous function f: IRn -+ Y is homotopic to a constant function, and that the same is true for any continuous function g : X -+ IRn .