If I am correct, a discrete subset of a topological space is defined to be a subset consisting of isolated points only. This is actually equivalent to that the subspace topology on the subset is discrete topology. There seems no restriction on the cardinality of a discrete subset, i.e. its cardinality can be any.
I was wondering if the following quote from wolfram is true and why?
Typically, a discrete set is either finite or countably infinite.
What kinds of topological spaces are "typical"?
Added: Is the following quote from the same link true
On any reasonable space, a finite set is discrete.
What kinds of topological spaces does "reasonable" mean?
Is discrete mathematics always under the setting of discrete sets wrt some topologies? In other words, is it a special case of topology theory? Or can it exist without topology?
Thanks and regards!