Intuitively, one would assume that the quotient space of a topological space under an equivalence relation would always be smaller than the original space. It turns out this is not remotely true. I'm specifically interested in quotient spaces of ℝ (under the standard topology).(adsbygoogle = window.adsbygoogle || []).push({});

We can easily make a quotient space of ℝ be homeomorphic to ℝ, for instance be gluing all the points in an interval into a single point. We can even glue infinitely many intervals into points, and still get a quotient space homeomorphic to ℝ. But my question is this: let us call an equivalence relation "nontrivial" if every equivalence class has at least two elements. Then does there exist a nontrivial equivalence relation on ℝ such that the quotient space is homeomorphic to ℝ?

Any help would be greatly appreciated.

Thank You in Advance.

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# Can a nontrivial quotient space of R be homeomorphic to R?

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