Defining homology groups directly from the topology
Both simplicial and singular homology theories rely on 'model objects', simplexes or simplicial complexes, to define the homology groups of a topological space.
I was wondering if there is a way to define homology groups directly from the open set structure of a space $(X,\mathcal{T})$. Here's a formal formulation of my question:
Is there a way to associate homology groups to bounded lattices with arbitrary joins so that $H_{n}(X)\simeq H_{n}(\mathcal{T})$, when seeing $\mathcal{T}$ as such a lattice ?
Of course this needs some hypothesis on $X$, if only to convene on a definition for $H_{n}(X)$. I don't mind if strong assumptions on $X$ (compact Hausdorff, triangulable) are needed.
Note that if the answer is positive, then $(X,\mathcal{T})$ and $(Y,\mathcal{T}')$ would have the same homology groups whenever $\mathcal{T}\simeq \mathcal{T'}$. But this is not surprising since in fact $\mathcal{T}\simeq \mathcal{T'}$ implies $X\simeq Y$ assuming both spaces are Hausdorff; see my previous question.
Here is the construction of a (co)homology theory for lattices which yields the Čech (co)homology in the case of topological spaces when applied to the lattice of open subsets. Recall that the Čech theory works with open coverings. An analogue of an open covering in a lattice $L$ is a subset $C$ of $L$ satisfying the property that the join of the elements of $C$ equals the unique maximal element of $L$. (The assumption is that $L$ is bounded and has arbitrary joins.)
I will refer to such $C$ as a covering of $L$. I say that a covering $C$ of $L$ refines a covering $C'$ of $L$ if for each $c\in C$ there exists $c'\in C'$ such that $c\le c'$. A refinement of $C'$ is such $C$ together with a map $f: C\to C'$
$$
f: c\mapsto c', \quad c\le c'.
$$
Given a covering $C$ of $L$ define its poset $P_C$ by taking meets of finite subsets of $C$. This poset yields a simplicial complex $X_C$, whose vertices are elements of $C$ and $c_0,...,c_n\in C$ define an $n$-simplex iff
$$
c_0 \wedge c_1 \wedge .... \wedge c_n\ne 0
$$
where $0$ is the least element of $L$.
The incidence relation between such simplices is the obvious one. Now, the Čech cohomology $H^*(L)$ (with coefficients in ${\mathbb Z}$) is defined as the direct limit
$$
\lim_{C} H^*(X_C)
$$
where the direct system is induced by the refinements of coverings $C$ of $L$. More precisely, if $(C,f)$ is a refinement of $C'$, we obtain a natural map of simplicial complexes
$$
\tilde f: X_C\to X_{C'}
$$
sending $c\in C$ to $f(c)\in C'$. Now, use pull-back map of the cohomology groups
$$
\tilde f^*: H^*(X_{C'})\to H^*(X_C).
$$
The direct limit of this direct system can be defined to be the Čech cohomology of $L$. One can define Čech homology similarly, using the homology and the associated inverse system instead of the direct one.