When is a quotient hausdorff

And nice example 2. I am wondering if the quotient space in the example is or homeomorphic to something known or useful in some applications? Or it is just an abstract construction useful as a counter-example only? I am not aware of any name for the quotient space nor of any direct use in applications.

It is an example showing the erratic behavior of the quotient near a hyperbolic fixed point of a dynamical system. See also Arnold's Cat Map for some nice pictures of a related system. Add a comment. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post.

MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem see for example, S. MacLane, Categories for Working Mathematicians. To every topological space this left adjoint associates a Hausdorff space called the largest Hausdorff quotient.

I doubt there is any actual reference for this. But this is not always the case. You can continue this for every ordinal number. It is interesting when we arrive at this stage, see my question about the nonhausdorff dimension. To augment very slightly Mariano's nice answer, Hausdorff quotients as opposed to surjections suffice. Their result applies to various similar separation properties.

Seminaire Bourbaki Exp. Sign up to join this community. The best answers are voted up and rise to the top. Anyway: Yes, the condition that the quotient map be open is necessary. You could post this as an answer to the new question math. Add a comment. Active Oldest Votes. Alex Youcis Alex Youcis 47k 8 8 gold badges silver badges bronze badges. Dylan Moreland Dylan Moreland PatrickR 2, 1 1 gold badge 13 13 silver badges 38 38 bronze badges.

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