(My original post starts here, and ends right before the Edit part. I am keeping it so that the comments and answer make sense, but what I am really interested in is what is in the Edit section.)

My experience has been so far mostly with finite-dimensional spaces. However, there is definitely a need to study infinite-dimensional spaces, even if one is mostly interested in finite-dimensional spaces, as the spaces of functions on finite-dimensional spaces are often infinite-dimensional.

Let us consider the infinite-dimensional polynomial ring $\mathbb{R}[x_1, \ldots, x_d]$. I am interested for instance in the set $\mathcal{I}_n$ of codimension $n$ ideals in that ring. Note that $\mathcal{I}_n$ can be thought of as a subset of the Grassmannian $\mathcal{G}_n$ of codimension $n$ subspaces of $\mathbb{R}[x_1, \ldots, x_d]$.

I am interested in the question of whether or not there is a natural topology on $\mathcal{I}_n$ which makes it into a smooth/continuous manifold. Maybe there is a manifold topology on $\mathcal{G}_n$ which makes $\mathcal{I}_n$ into a manifold?

I think that if one considers topologies related to the Zariski topology, then analogues of my questions above would have a negative answer, because the Hilbert scheme of degree $n$ points on affine $d$-dimensional space $\mathbb{A}^d$ could develop singularities if $d > 2$ (please correct me if I am writing something inaccurate or wrong).

Edit: here is what I am really interested in. In $\mathbb{R}^d$, you can consider the configuration space $C_n(\mathbb{R}^d)$ of $n$ distinct points in $\mathbb{R}^d$. I would like to define some kind of completion of $C_n(\mathbb{R}^d)$ which contains for instance degenerate configurations where $2$ or more of its points collide. The purpose is to enlarge the domain for instance for numerical approximations of the first order directional derivative of a function $f$ of $x_1, \ldots, x_d$ at some point $p$ and in some direction $v \in \mathbb{R}^d$ (i.e. $D_v(f)(p)$). For instance, one may approximate

$$D_v(f)(p) \simeq (f(p_1) - f(p_0))/h$$

where $p_0 = p$ and $p_1 = p + h v$ and $h$ is a small positive number. Note that if we enlarge the domain of the approximation so as to include the case where $h$ goes to $0$, so that $p_1$ and $p_0$ collide, we then get that the approximation is exactly $D_v(f)(p)$.

So I would like to enlarge the domain of numerical approximations so as to depend not just on finitely many distinct points, but to also include limiting configurations where two or more of these points collide.

There are two such approaches which come to my mind (well, that I know of) that could be relevant: the Fulton-Macpherson compactification and the Hilbert scheme of points.

So my question is really, which completion of $C_n(\mathbb{R}^d)$ is well suited for extending the domain of $n$ point numerical schemes? Another question is, has this been done? (Such questions were proposed to me by another Mathematician actually)

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