Questions about the Stone-Čech compactification A compactification of $X$ is a pair $( h, Y)$ where $Y$  is a compact space and $h\colon X ‎\to Y$ is an embedding such that $h(X)$  is dense in $Y$.


*

*According to definition of mentioned compactification, is the Stone-Čech compactification $\beta X$ Hausdorff?

*How can we introduce Stone-Čech compactification by means of ultrafilter?
 A: The Čech-Stone compactification $\beta X$ is defined only for Tikhonov spaces $X$ and is always Hausdorff. 
The ultrafilter construction of $\beta X$ technically applies only to discrete $X$; for an arbitrary Tikhonov space $X$ one uses maximal filters of zero sets, sometimes called $z$-ultrafilters. If $X$ is discrete, every subset of $X$ is a zero set, so $z$-ultrafilters are just ordinary ultrafilters. In this construction we let $\beta X$ be the set of $z$-ultrafilters on $X$. Let $\mathscr{Z}$ be the family of all zero sets in $X$. For each $Z\in\mathscr{Z}$ let $\hat Z=\{\mathscr{F}\in\beta X:Z\in\mathscr{F}\}$; then $\{\hat Z:Z\in\mathscr{Z}\}$ is a base for a topology on $\beta X$, and that topology turns out to be compact and Hausdorff.
Since $X$ is Tikhonov, for each $x\in X$ there is a unique $z$-ultrafilter $\mathscr{F}_x$ that converges to $X$, so we can define a map $$h:X\to\beta X:x\mapsto\mathscr{F}_x\;.$$ This map turns out to be an embedding of $X$ as a dense subset of $\beta X$.
That’s just a brief outline; for the details you’d be better off consulting a text.
For $T_1$ spaces $X$ there is something called the Wallman compactification $\omega X$, which is not necessarily Hausdorff, though if $X$ is normal it coincides with $\beta X$; it uses a similar construction, but with maximal filters of closed sets instead of maximal filters of zero sets.
