Are finite sets discrete by definition? Are finite sets countable by definition? I was first wondering about wether a discrete set was countable by definition, but this page states that that a discrete set has to be either finite or countable.
And although it's easy to come up with cases for sets that are


*

*finite and countable,

*infinite and countable,

*and infinite and uncountable,


it seems strange to me that a set could be finite and still uncountable.
I hope that this level of questions is allowed here, it's the first ime I post here, so I'm not sure about this yet.
 A: There is something confusing about the terminology "discrete". Discrete implies some sort of topology, some sort of sense to say whether two elements are close to one another. Discrete means that the elements are spaced.
If we talk about subsets of the real numbers, then finite sets are always discrete; and every discrete set is countable (or finite). However the rationals are not a discrete set, between two rationals you can always find another rational.
When I was a freshman we always assumed that discrete is interchangeable with countable (or finite), and I learned only later that this is a flawed concept. Discrete sets can be uncountable, in the broad context of mathematics, and finite sets can be made non-discrete as well in the broad context of mathematics.
What I do read from your question is whether or not countable includes finite. This depends on the context, and whether or not it would simplify things for us.
Sometimes we want to say that finite is countable, because it means things easier "A set is countable if and only if it is equipotent to a subset of the natural numbers" would include finite in the definition.
On the other hands, there are cases where the finite cases are irrelevant and serve as a point which we have to deal with separately, so defining countable to be countably infinite makes things easier. You deal with the finite case, and then you deal with the countable case.
In either case it is common in textbooks and papers to include a word on the meaning of countable. We could mean "at most countable" or "countably infinite". Usually one can deduce that from the context, if the other term appears.
That is, if we run into something like this:

If $A$ is countable, and $f\colon A\to\mathbb R$ is this and that, then the image of $f$ is at most countable.

would imply that countable means countably infinite, however is not necessarily the case and one should probably look for the definition in the preliminaries section as well.
A: A set $S$ is said to be countable iff there is an injection $f: S \rightarrowtail \Bbb N$.
It is therefore in bijective correspondence with a subset $S' \subseteq \Bbb N$ of $\Bbb N$; it is clear that all finite sets are of such form, hence all countable.
However, this is completely unrelated to the notion of discreteness.
A: To answer the question in the title, "no".  A topological space consisting of finitely many points with the indiscrete topology is finite but as its name suggests, not discrete (except in the trivial cases of zero or one points.)
