1. Riemann surfaces homework.
We let (X, dX), and (Y, dY), and (Z, dZ) be metric spaces.
(1) For any x16= x2 ∈ X, show that there exist r1, r2 > 0 so that the open balls Br1(x1) and Br2(x2) have empty intersection.
(2) Let f : X → Y be a function. We say that f is continuous at x0 ∈ X if for every > 0, there exists δ = δx0, > 0 such that dY(f (x0), f (x)) < whenever dX(x0, x) < δ. We say that f : X → Y is continuous if f is continuous at every point of X.
(a) Show that f is continuous if and only if for every open set V of Y, f−1(V ) is open in X.
(b) Use (a) to show that if f : X → Y and g : Y → Z are continuous functions, then g ◦ f : X → Z is also continuous.
(3) A metric space is connected if it cannot be represented as a union of two nonempty open subsets. Suppose (X, dX) is a connected metric space. Show that the only nonempty open and closed subset of X is X itself.
(4) Let f : X → Y be a continuous map with X compact. Show that f (X) is a compact subset of Y. Moreover when Y = R, show that f attains its maximum and minimum in X. (You can use the above exercise or prove it directly.)
(5) We say that X is locally path connected at x ∈ X if for every open neighborhood V of x, there exists an open neighborhood U of x contained in V such that U is path-connected. We say X is locally path connected if it is locally path connected at all x ∈ X. Show that a connected and locally path connected space X is path connected.
(6) A homeomorphism between metric spaces X and Y is a bijection h : X → Y such that both h and h−1are continuous. If there is a homeomorphism h maps X onto Y, we say that X and Y are homeomorphic. We say that X is topologically equivalent to Y if X is homeomorphic to Y. If X and Y are both open subsets of Rn and if h : X → Y is a homeomorphism such that both h and h−1 are differentiable, we say that X and Y are diffeomorphic.
(a) Show that topological equivalence between metric spaces is an equivalence rela- tion. (This equivalence can be extended to the class of all topological spaces.) (b) Similarly, diffeomorphisms define an equivalence relation between open sets of Rn. (You do not need to prove this.) Let Ba= {x ∈ Rn: kxk < a} with a > 0.
Show that the map
h(x) = ax
pa2− kxk2
defines a diffeomorphism of Baonto Rn. Here kxk denotes the Euclidean norm pP
i(xi)2 for x = (x1, · · · , xn).
(c) Give an example to show that a smooth bijective map need not be a diffeomor- phism.
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