I have been slowly getting more into math, but haven't gone into differential geometry or anything like that yet, so this question might be basic. Trying to get a deeper understanding of the Cosmological Principle, and why we assume it without question.
Physics/cosmology books make the assumption that the universe is isotropic, because (to summarize), "every observation we have made with telescopes show that the universe looks the same in all directions".
Because of that assumption, and the second assumption that the Earth/Sun aren't the center of the universe (the Copernican Principle), we make a final assumption that the universe must be homogeneous. Together creating the Cosmological Principle.
Has there been any evidence that these assumptions could be invalid? In terms of isotropy, do our observations that "the universe looks pretty much the same no matter where you look with a telescope" agree with a mathematical definition of isotropy? (Not sure what that, haven't dug much into algebraic/differential geometry yet). If it doesn't match exactly, what are some of the edge cases?