Let ℝ be a topological space with the cofinite topology, i.e. closed sets are finite. Then is ℝ×ℝ with the product topology homeomorphic to ℝ×ℝ with the cofinite topology? I don't even know where to start. Thank you in advance
Are these spaces with the cofinite topology homeomorphic
general-topology
Best Answer
They are not homeomorphic.
Within cofinite R, the only infinite closed set is R.
Within R×R, there are many infinite closed sets.
For example, R×K for every finite K that is a subset of R.