# Pure topological construction of the unit interval The challenge here is to construct this well known topology in pure topology methods without leaning on non-topology concepts like order or metric. For $x\in I$ there is an infinite binary representation: $x=0.a_1a_2,...$ where $a_i\in\{0,1\}$ and $x=\sum a_i2^i$. For $s=\{s_1,...,s_n\}$, define $[s]=\{x: x\in I , x=0.s_1s_2,..,s_n,....\}$ i.e., the numbers with $s$ in their prefix. $[s]=[\frac{m}{2^n},\frac{m+1}{2^n}]$ where $m=\sum_{i=1}^ns_i\cdot2^i$. This means $I=\bigcup_{|s|=n}[s]$ . $[s]$ generates all the standard closed sets of I and so we can define the open sets as their complements. Note: We need to take care of numbers with finite number of $a_is that can be represented with infinite zeros at the end or replace the last 1 with "0111111...". They can be handled by defining a quotient map s.t., two such numbers considered to be in the same equivalence task. But let's ignore this issue for clarity. ## Created 2026-06-03 12:18