If $C$ is a set of topological spaces then the usual topological product $\prod C$ is the topological space with underlying set $\prod C$ and basic open sets of the form $\prod U$, where $U(X)\subseteq X$ is open for all $X\in C$, and $U(X)\neq X$ for only finitely many $X$. In, e.g., forcing this is considered to be the 'finite support product' of the spaces, and one can easily generalise this idea:
If $I$ is an ideal on $C$, then let us define $\prod_IC$ as the topological space with underlying set $\prod C$ and basic open sets of the form $\prod U$, where $U(X)\subseteq X$ is open for all $X\in C$, and $\{X\in C\mid U(X)\neq X\}\in I$.
Is there a corpus of research on these generalised products from the topological point of view? Furthermore, given a sufficiently 'nice' ideal $I$ (for example, $I=[C]^{{<}\lambda}$ for some cardinal $\lambda$), is $\prod_IC$ the limit of some categorical diagram, or the solution to some universal property in the way that $\prod C$ is?