Recent content by dori1123

F is continuous if for every open subset U of Y, F^{-1}(U) is open in X. So I let U be an open subset in Y, and let (a,f) be in F^{-1}(U). Then F(a,f) is in U. then I am stuck...

I don't know how to show F(a,f)=af is continuous, can I get some hints please.

Let C([0,1]) be the collection of all complex-valued continuous functions on [0,1]. Define d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx for all f,g \in C([0,1]) C([0,1]) is an invariant metric space. Prove that C([0,1]) is a topological vector space

Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0. With a Euclidean circle centered at the origin with radius r, how can I find the hyperbolic radius by integrating \sqrt(E(u')^2+2Fu'v'+G(v')^2), what parametrized curve should I use?

Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0. How can I show that a Euclidean circle centered at the origin is a hyperbolic circle?

An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)\pir^2. But how is that obtained?

I've been reading a few proofs showing that a great circle is geodesic. Most of these proofs start with a parametrization and then show that it satisfies the differential equations of geodesics. The book that I have doesn't even give a proof. It just tells me that the great circles on the sphere...

Why stereographic projection preserves angles between curves but does not preserve area?

Show that if H is a normal subgroup of G of prime index p, then for all subgroups K of G, either (i) K is a subgroup of H, or (ii) G = HK and |K : K intersect H| = p.