1. hw2
(1) Let X and Y be metric spaces and X is compact.
(a) Suppose f : X → Y is continuous and surjective. Show that Y is also compact.
(b) Furthermore, if f is also injective, show that f is a homeomorphism.
(2) Find an example of an even dimensional smooth manifold which does not have any complex structure.
(3) Show that any open subset of a Riemann surface is again a Riemann surface.
(4) Let S2 = {(x, y, z) ∈ R3 : x2+ y2+ z2 = 1} be the unit sphere with the subspace topology induced from R3. Show that S2 has a structure of a Riemann surface.
(5) On C2\{0}, we define z ∼ w if z = λw for some λ 6= 0. Define P1 = (C2\{0})/ ∼ be the set of all equivalent classes of the above equivalence relation. Let π : C2\ {0} → P1 be the projection map. We say that U is open in P1 if and only if π−1(U ) is open in C2\ {0}. Show that P1 is isomorphic to S2 in the category of Riemann surfaces.
(6) Let X be a Riemann surface and U be any open subset of X. Show that the ring of holomorphic functions OX(U ) is independent of choice of complex atlas on X.
(7) Let X be a Riemann surface and (ϕ, U ) be a complex local chart on X. Use definition to show that ϕ ∈ OX(U ).
(8) Let f : X → Y be a continuous map between metric spaces. For any open set V of Y, define
f∗ : C(V ) → C(f−1(V )), h 7→ h ◦ f.
Show that f∗ is a ring homomorphism. Furthermore, if both X and Y are Riemann surfaces, show that f∗(OY(V )) is a well-defined subring of C(f−1(V )). Here C(X) denotes the algebra of complex-valued continuous functions on a metric space X.
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