Shrinking good coordinate systems associated to Kuranishi structures

Abstract

The notion of good coordinate system was introduced by Fukaya and Ono in [FOn] in their construction of virtual fundamental chain via Kuranishi structure which was also introduced therein. This notion was further clarified in [FOOO1] in some detail. In those papers no explicit ambient space was used and hence the process of gluing local Kuranishi charts in the given good coordinate system was not discussed there. In our more recent writing [FOOO2, FOOO3], we use an ambient space obtained by gluing the Kuranishi charts. In this note we prove in detail that we can always shrink the given good coordinate system so that the resulting ‘ambient space’ becomes Hausdorff. This note is self-contained and uses only standard facts in general topology.