# How to express, say, the Hausdorff property of a topological space using categories. (And a request for general advise in such practices)

I am new to Category theory and for the sake of the practice, I am interested in revisiting and expressing those concepts that I am familiar with --however basic--, in the language of categories. While categories deal with subtle things like homology quite gently, the use of them in basic undergraduate math, seems unnecessary and usually is avoided. Here is one example:

In elementary number theory, the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) can be seen as products and coproducts in the category which its objects are natural numbers and its morphisms are build out of divisibility expressions:

$$dom f |cod f \iff dom f\stackrel{f}\longrightarrow cod f$$ While there is no terminal object in this category, 1 is its initial object and we can also characterize primes, unites and factorization in terms of arrows.

Now here is my specific question:

How to formulate compactness and various separation axioms, particularly Hausdorff property, in terms of arrows in Top.

The second part of my question is a request for reading materials (apart from Maclane's), which would include such things and more. Thank you.

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Compactness is easy enough to formulate: $X$ is compact if and only if for any filtered jointly surjective family of open immersions into a space $X$, just one of those open immersions is already surjective (hence a homeomorphism). If you think about it this is just a fancy way of saying that every open cover has a finite subcover. –  Zhen Lin Oct 26 '12 at 22:58
Brilliant. Thank you. –  Hooman Oct 26 '12 at 23:00
It is also well-known that $X$ is Hausdorff if and only if the diagonal map $X \to X \times X$ is a closed immersion. –  Zhen Lin Oct 26 '12 at 23:01
@ZhenLin, it sounds as if there's a general-topology concept of "immersion" that I'm not familiar with. I know the one for smooth manifolds, but not general topological spaces. What is this? Is it, for example, a map which is locally an embedding? –  Hew Wolff Oct 27 '12 at 2:03
I'm just thinking of an injective continuous map that is a homeomorphism onto its image. –  Zhen Lin Oct 27 '12 at 6:35