$\newcommand\seq[1]{\langle#1\rangle}$A large number of important topological results require simplicial-algebraic machinery (or comparable) to prove. This machinery is ingenious, impressively so even, yet I cannot shake the feeling that a pointfree approach might be more direct.
If we take the unit interval as example then we could specify this recursively, perhaps a bit cryptically, as follows:
$A_{\seq{}}\ \ $ (the pointfree element representing $[0,1]$)
$A_s\longrightarrow A_{s*0}\approx A_{s*1}\approx A_{s*2}\,$ for $s\in\{0,1,2\}^*\ \ $ (a pointfree interval subdivides in three)
$A_t\approx A_{S(t)}$ where $S(t)$ denotes the successor of $t$ in the usual lexicographical ordering on $\{0,1,2\}^n$, for $t\in\{0,1,2\}^n, t\ne \seq{2\ldots2}\ \ $ (touching is preserved).
The symbol $\approx$ can be read as "touches" or "non-empty overlap", its negation is the symbol $\#$ ("apart" or "disjunct").
The unit interval can now be identified with $\sigma_3=\{0,1,2\}^{\mathbb N}$ equipped with the equivalence relation $\equiv_{\mathbb{R}}$ given by:
- $\alpha\equiv_{\mathbb{R}}\beta$ iff $\forall n\in\mathbb{N}[A_{\alpha\mid n}\approx A_{\beta\mid n}]\ \ $ where $\alpha\mid n$ denotes the string of the first $n$ values of $\alpha$.
In fact the $\equiv_{\mathbb{R}}$-quotient topology on $\sigma_3$ (derived from the Baire topology) is homeomorphic with $[0,1]$. This is understood also by seeing the whole thing as the ternary real numbers in $[0,1]$, where 'ternary' is the 'decimal' expansion with base $3$.
Generalization of the above yields what I have dubbed 'apartness topology' in my 1996 PhD thesis (the terminology has later been adopted by others but without the pointfree approach). Apartness topology with the pointfree flavour gives an elegant way in intuitionistic mathematics to tackle all 'effective' topologies on Baire space, which can be shown to coincide with the quotient topologies of a $\Pi^1_0$ equivalence relation on Baire space.
My question more specifically is this:
(i) has anyone considered constructing an algebraic structure directly from a pointfree definition of a topological space in such a way that the simplicial approach can be bypassed?
(ii) has such an algebraic structure been directly tied to a recursive definition similar to the one above?
One will observe that the recursion rule above is tied in with simplicial subdivision, but from thereon it gets above my head. Any references as well as comments will be appreciated.
