Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes, then what is the cardinality of $\xi$ ?



A question on cofinite topology.
You should mean $\{x\}=\bigcap\xi$, and the answer is clearly yes, since we can take $\xi$ equal to the set of all open sets containing $x$. Any point $y$ other than $x$ is excluded in this intersection by the open set $X-\{y\}$. The cardinality of this $\xi$ is the same as the number of finite subsets of $X$, which is equinumerous with $X$. And the same will be true of any $\xi$ having your property, since the points other than $x$ must get excluded by elements of $\xi$, but only finitely many at a time, and so the cardinality of $\xi$ must be the same as $X$. 


(This argument uses the axiom of choice, in order to know that the collection of finite subsets of an infinite set is equinumerous with that set. In $\neg$AC worlds, the situation is more complicated.) 


