Characterizing uncountable connected topological spaces 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 ?
 A: 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.


*

*The class of connected metric spaces (as you stated)

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

*The class of non-discrete complete metric spaces

*The class of connected Hausdorff compact spaces

*The class of uncountable spaces

*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. 
A: 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.
