Hello, congratulations for attempting to build Sage
from source on Cygwin and thanks for reporting on
the build failure.

Which version of Sage were you trying to build?
Did you try SageMath 9.4 or SageMath 9.5.rc1?

Fixing the Python 3.9 build failure you described is tracked at:

  Sage Trac ticket #33079
  Fix python3 build failure on Cygwin
  https://trac.sagemath.org/ticket/33079

In general, Sage's build mechanism tries to use the ambient
system Python, if available, to avoid building its own Python.
However, the Python 3.9 in Cygwin, currently provided by the
Cygwin package "python39-3.9.9-1", does not work for that
purpose. That issue is discussed at:

  Sage Trac ticket #33078
  Update package info for system python3 on Cygwin
  https://trac.sagemath.org/ticket/33078

There is a newer version which can serve to build Sage upon;
that newer version is currently marked as a test version, so that
it does not get installed by default when you upgrade your Cygwin
packages. You can however choose to upgrade to it, as follows.

- Close all Cygwin windows.
- Open Cygwin's "setup.exe" installer, search for "python39",
  and select the version "python39-3.9.9-3 (TEST)".

Please try again building SageMath 9.5.rc1 after changing that.
Let us know if that works. Currently it does not work for me,
but I hope it works for you, and if not I hope we gather enough
data on the failure to figure out how to solve it.   --Samuel

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