Discrete subsets of real space must be countable BUT other discrete sets can be uncountable? What are some examples for the latter claim?
I am comfortable with the fact that a discrete subset of $R$ is countable. And I think we might have some complicated space in which the discrete subsets can be uncountable. Am I correct?
 A: Just take an uncountable set equipped with the discrete topology. For instance, you could take the metric space $({\Bbb R}, d)$, where $d$ is the discrete metric
$$
d(x,y)= \begin{cases}
1 & \text{if $x \not= y$}\\
0 & \text{if $x = y$}
\end{cases}
$$
A: With regard to the OP's comment, it's probably more meaningful to look for topological spaces, which contain uncountable discrete subsets, but fullfill other countability conditions, eg., which are separable or Lindelof. Such a space cannot be metrizable, though.
The Stone-Cech compactification of the integers would be a standard example:
It is compact, hence Lindelof. Since the integers are dense in it, it is separable. By a somewhat more complicated argument, it contains an uncountable, discrete subset (even of cardinality c (see, for instance, Engelking's book, 3.6.18).
An easier example is the one-point-compactification of an uncountable, discrete space. But this is not separable.
If you are looking for further examples, you can search for "uncountable spread".
