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Largest Hausdorff quotient

Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$?

If this is too much to ask, how about a left adjoint to the restriction to compact spaces ${\mathbf{CptHaus}}\to{\mathbf{CptTop}}$?

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    $\begingroup$ This is a duplicate: mathoverflow.net/questions/78175/largest-hausdorff-quotient $\endgroup$
    – Theo Buehler
    Commented Dec 30, 2011 at 8:44
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    $\begingroup$ Yes, there is such an object. The idea for the definition comes from the universal mapping property encoded in the definition of adjoint. I'd call this functor the "maximal Hausdorff quotient" of a space. This is a good homework problem in a point-set topology course. Given a topological space $X$, consider all equivalence relations on $X$ with Hausdorff quotient, and take the intersection of all such equivalence relations. $\endgroup$
    – Ryan Budney
    Commented Dec 30, 2011 at 8:54

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