# Universal Hausdorff Space [duplicate]

Possible Duplicate:
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 ${\mathbf{CptHaus}}\to{\mathbf{CptTop}}$?

-

## marked as duplicate by Ryan Budney, Gjergji Zaimi, Qiaochu Yuan, Dmitri Pavlov, Andreas BlassDec 30 '11 at 15:36

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. – Ryan Budney Dec 30 '11 at 8:54