Shrinking good coordinate systems associated to Kuranishi structures

TL;DR

This paper demonstrates how to shrink good coordinate systems associated with Kuranishi structures to ensure the ambient space is Hausdorff, providing a detailed, self-contained topological proof relevant to virtual fundamental chain constructions.

Contribution

It introduces a method to shrink good coordinate systems to achieve Hausdorff ambient spaces, clarifying the gluing process of Kuranishi charts in this context.

Findings

Ambient space can be made Hausdorff by shrinking the coordinate system.

The process uses standard topology techniques.

The result simplifies the construction of virtual fundamental chains.

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.