Is a connected set always an uncountably infinite set?

Question

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?

Answer 1

Consider the space $X = \{a, b\}$ equipped with the trivial topology (i.e. the only open sets are $X$ and $\varnothing$). Then, the space is connected, but there are only a finite number of points in this example.

Answer 2

First of all, $\emptyset$ is connected. And so is every singleton.

On the other hand, it is true that every connected metric space with more than one point is uncountable. But there are countable connected topological spaces.

Answer 3

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).