Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind of subset, namely one that is the product of its cross sections. Perhaps a pipe, or a tube, or a cylinder, or a cube, or a rectangle, or a prism. I can't recall. Is there a commonly accepted terminology for this kind of subset?
What is the terminology for a subset of a product of sets that is the product of its cross-sections?
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1In this generality, $S$ is just the product of $(S_x \mid x \in X)$. In more specific instances, there may be other common names as well. Suppose for example that $O$ is an open disc in some (metric) space and that $S_x = O$ for all $x \in X$ with $X$ finite. Then $\prod_{x \in X} S_x$ is a generalization of a cylinder in $R^{\mathrm{card}(x)}$ and thus I could easily see someone call products like this cylinders. – Stefan Mesken Nov 22 '18 at 15:00