Characterizing uncountable connected topological spaces

Question

We know that if $X$ is a connected metric space with more than one point , then $X$ is uncountable ; can we characterize those connected topological spaces for which more than one point implies uncountability ?

Answer 1

As this question is posed, it does not make much sense.

We can list some classes of topological spaces for which the property $$\mbox{if $X$ has at least two points, then X is uncountable}$$ holds. Let's make some examples.

1. The class of connected metric spaces (as you stated)

2. The class of spaces with one point (in this case the property is vacuously satisfied)

3. The class of non-discrete complete metric spaces

4. The class of connected Hausdorff compact spaces

5. The class of uncountable spaces

6. Any subclass of the preceding classes

As you can see, a lot of classes of spaces satisfy this property, but I hardly see how one can characterize them all.

Answer 2

It is even true that every functionally Hausdorff connected space with at least two points has size at least continuum. In turn, every $T_3$ connected space with at least two points is uncountable (a countable one would be $T_4$ and hence functionally Hausdorff). On the other hand there exists a countably infinite connected Hausdorff (even Urysohn) space.