Explicit homeomorphism between an Infinite Cone and a one sheet hyperboloid

Question

An infinite cone is given by the equation

$$B = \{(x,y,z) ∈ \mathbb R^3 : x^2 + y^2 −z^2 = 0\} $$

And a one sheeted hyperboliod is given by

$$C = \{(x,y,z) ∈ \mathbb R^3 : x^2 + y^2 −z^2 = 1\} $$

Are these two surfaces homeomorphic? And if yes, what are their explicit homeomorphisims?

I know both of these spaces are not compact, however I do not know how to check for other properties such as connectedness which are preserved under homeomorphisims.

Answer 1

HINT: What happens if you remove the origin from the cone? Well, if they were homeomorphic, you'd remove the corresponding point from the hyperboloid. Now what?

Answer 2

Look at the intersection of each of these surfaces with the plane $z=1.$

By moving that plane downward from $z=1$ to $z=0,$ you can shrink that loop on the cone down to a point. But there's no way to do that on the hyperboloid; it's not simply connected.