the Stone-Čech compactification by using ultrafilters If $X$ is discrete, one can construct $\beta X$ as the set of all ultrafilters on $X$.

But which kind of topology must we use in the above sentence? 

How can we define the Stone–Čech compactification by using ultrafilters? What is the form of the topology in terms of ultrafilters?

For each $A\subseteq X$ let $\widehat A=\{p\in\beta X:A\in p\}$; then $\{\widehat A:A\subseteq X\}$ is a base for the topology of $\beta X$. Note that if $x\in X$, then 
$$\widehat{\{x\}}=\left\{p\in\beta X:\{x\}\in p\right\}=\{p_x\}$$
where $p_x$ is the principal ultrafilter over $x$: $p_x$ is an isolated point in $\beta X$. This is why we identify $p_x$ with $x$ and say that $X\subseteq\beta X$, when really it’s $\{p_x:x\in X\}$ that is the subset of $\beta X$ corresponding to $X$.
