Cardinality of a discrete subset

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.

1. 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"?

2. 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?

3. 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!

For 1, spaces in which the discrete subsets are at most countable, and these are called spaces with "countable spread" in topology. Here, the spread $s(X)$ of a space $X$ is defined as the supremum of the cardinalities of all discrete subspaces of $X$, where by convention a finite supremum is rounded up to $\aleph_0$ (only infinite cardinals are used), also because every infinite Hausdorff space has a countable discrete subset (so spaces with a finite spread would be "pathological" non-Hausdorff spaces, or finite to begin with).

If a space is second countable, then every subspace is second countable too, and a discrete second countable space is at most countable, so a second countable space has countable spread. But this argument can be repeated for other classes of spaces: if every subspace of $X$ is separable ($X$ is then called hereditarily separable) or every subspace of $X$ is Lindelöf ($X$ is then called hereditarily Lindelöf) then $X$ has countable spread too (as a Lindelöf discrete space or separable discrete space both must be countable). For metrizable spaces, countable spread is equivalent to being separable, or Lindelöf, or second countable. See my post on topology atlas, but in general this need not be the case. But the Wolfram quote maybe comes from the fact that a lot of mathematics is done in separable metrizable spaces, like the Euclidean spaces.

An example of a separable compact space that does not have countable spread is $\beta(\omega)$ or $[0,1]^{\omega_1}$.

As to 2, the property that all finite subsets are discrete is equivalent to being $T_1$ (defined either as all singleton sets are closed, or for every $x \neq y$ in $X$, there are open sets $U$ and $V$ such that $x \in U, y \notin U$ and $y \in V, x \notin V$). This already follows from considering subsets of 2 points.

As to 3, adding a discrete topology to a set doesn't make it any more topological, as all functions on it are continuous, there are no non-trivial convergent seuqneces or nets, etc. So a discrete topology adds no information. It's true, for example, that any group can always be given a discrete topology and then it's a topological group (the group operations are continuous), but if we apply theorems from the general theory of topological groups, we cannot prove anything new that we couldn't prove by just plain algebra/group theory. The same holds for other types of (finite or not) structures in discrete mathematics: discrete here is opposite to "continuous", one could say: we do not consider topological or analytical structure, but just the structure as a set. The discrete topology is as informative as no topology in this case....

Yes. Put the discrete topology on [0,1]. This "typically" means in a separable space.

For the added Question $2$, here is an unreasonable space. Underlying set: the reals, or the integers from $1$ to $10$, or indeed any non-empty set. Open sets: the empty set and the whole space, that's all! This is a topology, usually called the indiscrete topology, or the trivial topology.

If $W$ is such a space with more than $1$ element with the trivial topology, then no non-empty subset of $W$ is discrete.

There are quite less extreme examples. One surprisingly important one (it has some applications in Theoretical Computer Science) is the Sierpinski Space. It only has two elements, say $0$ and $1$. The open sets are everything except $\{0\}$. The finite set $\{0,1\}$ is not discrete, since any open neighbourhood of $0$ is the whole space.

As to Question $3$, almost all of discrete mathematics is unconnected with general topology. Algebraic topology is another matter, here there are deep and fruitful connections.