Prove that the double cone x y z belongs to R3 x2 y2 z2 i

Prove that the (double) cone {(x, y, z) belongs to R^3 : x^2 + y^2 = z^2} is path-connected.

Solution

Proof:

If (p,q,r) is a point of the double cone D, we can define f:[0,1] --> D, f(t) = (tp, tq, tr).

As (tp)^2 + (tq)^2 = t^2(p^2 + r^2) = tq^2 (as (p,q,r) is in D) = (tq)^2, we have that

all points f(t) are indeed in D, and this defines a path from (0,0,0) to (p,q,r). Continuity is obvious.
So all points are connected to the origin by a path. And so we can find a path
between any 2 points of D, by going via the origin and combining the paths in the usual way.

All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy

h_t(x,s) = (x, (1t)s).The cone is used in topology precisely because it embeds a space as a subspace of a contractible space.

When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone CX can be visualized as the collection of lines joining every point ofX to a single point