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Cantor Set (C) homeomorphic to C x C

secretluna

New member
Dec 7, 2021
1
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.