How to prove that this is a lattice?

Question

Given the poset $(X,\subseteq)$ with $X=\{T\subseteq\mathbb{N}\ |\ T$ is finite or $\mathbb{N}\setminus T$ is finite}

How can I prove, that for some arbitrary $X_1, X_2\subseteq X$ there exists the supremum and the infimum?

Answer 1

First prove that $X$ is closed to the union and intersection by considering all combinations:

1) If $X_1$ finite and $X_2$ finite then $X_1\cup X_2$ is finite and $X_1\cap X_2$ is finite

2) If $X_1$ finite and $\mathbb N \setminus X_2$ finite then $\mathbb N \setminus (X_1\cup X_2)\subseteq \mathbb N \setminus X_2$ is finite and $X_1\cap X_2$ is finite

3) If $\mathbb N \setminus X_1$ finite and $\mathbb N \setminus X_2$ finite then $\mathbb N \setminus (X_1\cup X_2)=(\mathbb N \setminus X_1)\cap(\mathbb N \setminus X_2)$ is finite and $\mathbb N \setminus (X_1\cap X_2)=(\mathbb N \setminus X_1)\cup(\mathbb N \setminus X_2)$ is finite

Then the union is upper bound:

$X_1\subseteq X_1\cup X_2$ and $X_2\subseteq X_1\cup X_2$

Then the union is the least upper bound: If $Y$ us another upper bound then:

$X_1\subseteq Y$ and $X_2\subseteq Y$, so $X_1\cup X_2\subseteq Y$

Proof of the intersection being greatest lower bound works similarly (by duality).