Currently I am trying to prove that 1/3-Cantor set is homeomorphic to $\{0,1\}^\mathbb{N}$ I tried personal approach but after a while, when I got stuck I went with research and found this and especially this. So in the second post everything seems clear to me, to the point where author intersects C-set with interval $[h^*,h^*+(2.3)^{-j})$. I see from where it came, but I am not quite sure why it is 2.3. I mean, I see that we are looking at bare construction of Cantor set and we start "building" its parts starting from point 0, but I don't see why it has to be by 2,3. Or maybe it's some mistake in notation... idk.
Still, despite it, I also tried to check his case for a=1, which is meant to be similar. And I would be grateful if someone can check it and say if whether it's wrong or not.
So:
For $ a=1, j \in N, h \in H(j,1)$ we have:
$ \{x\in C:\forall i\leq j (x_i=h(i)\}= C\cap (h^*-2.3^{-j},h^*] $ as we go from point 1.
Now h* is either 1 or lower endpoint of $(h^*,h^*+3^{-j+1})$, which were deleted due to construction of C. Moreover because a=1 we have
$ \{x\in C:\forall i\leq j (x_i=h(i)\}= C\cap (h^*-2.3^{-j},h^*+3^{-j+1}) $ which is open in C.
It may be trivial, but still, I will be grateful if someone can look at it and point out my mistakes. Thanks for help!