In the lecture, we introduced the Stone–Čech compactification via ultrafilters. More concretely, we defined $\beta X = \{\mathfrak{U}|\mathfrak{U}$ ultrafilter on $X\}$.
This is possible for $X$ being a discrete set. I am wondering why this is necessary. What can cause troubles if X is not discrete/where is the discreteness needed so that βX is the space of ultrafilters?
Intuitively, a discrete set is "small", ensuring that the map from $X \rightarrow \beta X$ is injective. We have proved that there are at least $2^{2^\kappa}$ ultrafilters for $\kappa$ being the cardinality of $X$, $\kappa ≥ \omega = |\mathbb{N}|$ but this should always be injective.
The Stone topology on $\beta X$ makes the principal ultrafilters $p_x$ isolated points for every $x \in X$, so if the canonical map $x \mapsto p_x$ is continuous (one of the properties of the Stone-Cech compactification), it forces $X$ to be discrete.