The epimorphisms of the category Haus are exactly the image-dense morphisms

TL;DR

This paper proves that in the category of Hausdorff spaces, epimorphisms are precisely the image-dense morphisms, using the Hausdorff quotient to establish the necessary and sufficient conditions.

Contribution

It provides a detailed proof characterizing epimorphisms in the category of Hausdorff spaces as exactly the image-dense morphisms, including a novel use of the Hausdorff quotient.

Findings

Epimorphisms in Hausdorff spaces are exactly the image-dense morphisms.

The Hausdorff quotient helps characterize when a space is Hausdorff.

Non-image-dense morphisms have Hausdorff quotients with at least two points.

Abstract

This document presents the proof that the epimorphisms of the category of Hausdorff spaces are exactly the image dense morphisms. While it is a classical result; its proof is difficult to find in internet. Consequently, I decided to write the proof on my way and to share my knowledge freely on Arxiv. This document can interest any student in Pure Mathematics in the topics of topology and category theory. The image-dense morphisms are clearly epimorphisms. To etablish the converse, I use an analogous to the Kolmogorov quotient that I call Hausdorff quotient. Thanks to this Hausdorff quotient, one can prove that the Haus-category is a reflective subcategory of the category of topological spaces. As it is also a full subcategory, we deduce a necessary and sufficient condition for a topological space to be Hausdorff. We use this condition to show that if a morphism f:A->B is not image-dense, then the Hausdorff quotient of the quotient B/cl(f(A)) has at least two points. Moreover, we show that if f is an epimorphism, the Hausdorff quotient of the quotient B/cl(f(A)) has at most one point. We deduce the result.