The following series of questions gives me no rest.
Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the system of polynomial equations.
What are the topological properties of $\mathbb{\widetilde{R}}^n$? Especially, what are singular (co)homologies of this space? Is it still contractible as $\mathbb{R}^n$ with usual topology?
We may go further and consider subspaces of $\mathbb{\widetilde{R}}^n$ with induced topology. For example, let
$\iota:M_g\rightarrow\mathbb{R}^3$
be some embedding of surface of genus $g$ into $\mathbb{R}^3$. Denote $M^{\iota}_g$ the space $\iota(M_g)$ with topology induced from $\mathbb{\widetilde{R}}^3$. First, does it depend on the embedding $\iota$? And second what are singular (co)homologies of $M^{\iota}_g$?
I would like to emphasize that I am least interested in the answer (as though it is also very interesting), but rather in technique of calculation of (co)homologies of such spaces. For example, $M^{\iota}_g$ is no more CW-complex and calculations via cellular complex are impossible.
In a sense, we can go in the opposite direction and consider $A\subset\mathbb{R}^n$ defined by a system of polynomials. As I understand in this situation $A$ is called algebraic variety over $\mathbb{R}$. The natural topology on $A$ is Zariski topology but we want to act like perverts, so we induce the topology on $A$ from usual $\mathbb{R}^n$. What can be said about (co)homologies (and more important about tools of their calculation) of these spaces? Especially, is it possible to say something about (co)homologies looking only on polynomials defining $A$ (I doubt, of course)?
It seems that foundamental algebraic topology books (Fomenko-Fuks, Hatcher) are based on standard topology of $\mathbb{R}$, so when I thought about some monstrous (but kind of natural in some cases) topologies I realized that I know nothing about them. So, any references are welcome.
Edit. Singular chains are maps $f: \Delta^k \rightarrow\mathbb{\widetilde{R}}^n$, where $k$ is dimension of the corresponding simplex, however, if this simplex has induced topology from $\mathbb{R}^k$ (not $\mathbb{\widetilde{R}}^k$) then obviously $C_i(\mathbb{R}^n)\subset C_i(\mathbb{\widetilde{R}}^n)$ (because Zariski topology is included in usual topology). However, at the moment I cannot give an example of mapping $f:\Delta^1 \rightarrow \mathbb{R}^2$ which would be continitous in Zariski sense and discontinious in usual sense, I think this is also interesting question.
Another thought in this direction, what will happen if one considers $\widetilde{\Delta}^k$ (Zariski topology on simplex) instead of $\Delta^k$. It leads to something that may be reasonably called Zariski-singular (co)homology theory. I just made it all up, so I would be all the way greatful to somebody who can review and say, does it all make any sense?
