Cantor Set (C) homeomorphic to C x C

secretluna

New member
Hi all,

I have proved that the function $$\displaystyle f: C \to \prod_{k = 1}^{\infty} \{0,1\}$$ is a homeomorphism (where $$\displaystyle C \text{ is the Cantor Set}$$). However, I am confused about how this can be applied to prove that $$\displaystyle C \times C \to C$$ is a homeomorphism.

My idea was to show that the projection map $$\displaystyle p_x((x_1, x_2)) = x_1$$ for $$\displaystyle (x_1, x_2) = x \in C \times C$$ is a continuous bijection between a compact space and a Hausdorff space. Since the Cantor Set is compact, it follows that its product is also compact. Since the Cantor Set is homeomorphic to a Hausdorff space (shown with the proof that $$\displaystyle f: C \to \prod_{k = 1}^{\infty} \{0,1\}$$ is a homeomorphism), it follows that the Cantor set is Hausdorff. It is my understanding that projection maps are continuous and surjective, so I would have to show that the projection is injective, but I am having trouble, which makes me think that I am on the wrong track.