Given two infinite sets $A$, $B$ of natural numbers, write $A\preceq B$ if $B\setminus A$ is a finite set. Define the equivalence relation $A\sim B$ if $A\preceq B$ and $B\preceq A$, and let $\partial\mathbb{N}$ be the set of equivalence classes of infinite sets under this equivalence relation. Write $[A]$ for the equivalence class of $A$.


Now define a topology on the disjoint union $\overline{\mathbb{N}}=\mathbb{N}\cup\partial\mathbb{N}$ as follows: A set $U\subseteq\overline{\mathbb{N}}$ is open if and only if, for every $[A]\in U\cap\partial\mathbb{N}$, $[B]\in U$ whenever $B\prec A$, and moreover $A'\subset U$ for *some* $A'\sim A$.



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> Is this a known topology? Does it have a name?
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It is not hard to see that $\overline{\mathbb{N}}$ is compact (hence the question title): For any neighbourhood of $[\mathbb{N}]$ is the entire space minus a finite subset of $\mathbb{N}$. In particular, it is very much a non-Hausdorff space. Also, $\overline{\mathbb{N}}$ contains $\mathbb{N}$ as an open subset with the discrete topology on the latter. The subspace $\partial\mathbb{N}$ is indeed the boundary of $\mathbb{N}$ in this topology (hence my chosen notation), and it is an [Alexandrov space](https://mathoverflow.net/questions/18393/what-do-you-call-a-topology-that-is-closed-under-arbitrary-intersections).


I came up with this topology while thinking about sequences, subsequences, and their limits. It seems rather natural, so I don't think I am the first one to ever think of it.



Is this a known compactification of the natural numbers?
Your set $\partial\mathbb{N}$ is also intensely studied in set theory and known as P(ω)/Fin. What you have done is mod out by the ideal of finite sets. People study more general properties P(X)/I, taking the quotient by many other ideals (or by an arbitrary ideal). P(X)/I is a Boolean algebra, and many forcing arguments can be viewed as forcing with this Boolean algebra. The topological properties are very much used in that forcing context, since the generic filters are exactly those containing elements from every ground model dense set. The finite sets become equivalent to the point [emptyset] in this algebra.


Perhaps Lusin was the first to study P(ω)/Fin seriously, and found the phenomenon of Lusin gaps. A gap in P(ω)/Fin is a cut in the order, where the left side increases and the right side decreases, and everything on the left is below everything on the right, with respect to almost-inclusion. Lusin found gaps of various types, including ones with uncountable cofinality.


A particularly interesting case is P(ω1)/I, where I is the ideal of non-stationary sets, and many set theoretic hypotheses, some engaging with large cardinals, interact with the topological properties of that situation. 


A few quick examples: 


* [Farah, How many Boolean algebras P(ω)/I are there?](http://www.math.yorku.ca/~ifarah/Ftp/ch.pdf)
* [Farah, Luzin gaps](http://www.jstor.org/stable/3844945)
* [Just and Krawczyk, Isomorphisms of P(ω)/I](http://www.jstor.org/stable/1999489)
* [Jech, Precipitous Ideals](http://www.jstor.org/stable/2273349)
* [Matsubara, Saturated Ideals and the SCH](http://www.jstor.org/stable/2275442)
* There are many others, including probably some good introductory surveys.


