In a lecture of real analysis (this course is about Lebesgue measure) the lecture said:
For a set $X$ - $P(X)$ (the power set) is a compele lattice: For every $S\subseteq P(X)$ there exist $\cup S=\cup_{A\in S}A$ and $\cap > S=\cap_{A\in S}A$
I read in Wikipedia that a lattice is called complete if it have a supremum and an infimum, I don't know exactly what a lattice is, but I do know that $(P(X),\subset)$ is partially ordered set. However, It doesn't seem that $\cap S\leq S$ and that $S\leq\cup S$.
Can someone please explain (in simple words) the meaning of a complete lattice, why does $\cup S,\cap S$ are called the supremum and the infimum ? As far as I see it is not true that $\cap S\leq S$ and that $S\leq\cup S$ if we treat $(P(X),\subset)$ as a partially ordered set.
[note: I am sorry for the many typos, I am working on the PC farm, and the Lyx here doesn't have a spell checker]