TITLE: Is a connected set always an uncountably infinite set?
QUESTION [6 upvotes]: I'm trying to understand the concept of a connected set. The classic example which is presented is that of $\Bbb R$ or any interval of $\Bbb R$ with the usual topology.
Moreover, I heard that to a certain extent connected sets can be considered opposite to discrete sets. They are sometimes indicated as representing the idea of a continuum. So, intuitively, shouldn't they always be uncountable sets?

REPLY [5 votes]: Well, there's the empty set, as others have noted. Also, every singleton is connected. But you can say a bit more if you assume some separation.

Any finite $T_1$ space is discrete, so it is connected if and only if it has only one point, so if you assume $T_1$, there are no nontrivial connected finite sets or spaces.
There are also connected $T_0$ spaces of any size you want (including any finite): if you take a nonempty set $X$ and fix $x\in X$, then if you declare open sets to be exactly those containing $x$, then the resulting topology is $T_0$ and connected (all points except for $x$ are closed).
On any infinite set (possibly countable), the cofinite topology is $T_1$ and connected.
There are connected countably infinite Hausdorff spaces. (and even $T_{2\frac{1}{2}}$-spaces).
A connected $T_4$ space with more than one point is necessarily uncountable. This follows easily from Urysohn's lemma (in fact, it follows that a nontrivial connected $T_4$ space has at least the cardinality of the continuum).
One can show that a countable $T_3$ space is $T_4$, so by the preceding point, a connected $T_3$ space cannot be countably infinite.

Incidentally, in topology, a continuum is a kind of connected set (namely one which is compact Hausdorff, or even metrisable, depending on the author).