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In this paper, the (infinite) direct product of fields is investigated. Inparticular, the finiteness of a given set is characterized in terms of somering-theoretic observations. Next, a certain localization (whose multiplicativeset formed by cofinite sets) of the direct product of fields is studied.Finally, it is shown that every set $X$ can be made into a separated scheme,and this scheme is an affine scheme if and only if $X$ is a finite set.

On the direct product of fields with an application