Prove that function is homeomorphism.

Prove that
$$
f: \prod\limits_{1}^{\infty} ( \{0,2 \}, \mathcal{T} _{\delta}) \to ([0,1], \mathcal{T}_{e}):\{n_i \} \mapsto \sum_{i=1}^{\infty} \frac{n_i}{3^i}
$$
is homeomorphism, and image of $f$ is Cantor set.

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Since you’ve given no indication of where you’re having trouble, I’ll give a couple of pointers to previous answers that should at least get you started. Here is a proof that the map is injective. This answer contains most of what you need to prove that the range of $f$ is the familiar middle-thirds Cantor set.