Connected and regular Hausdorff with at least two points implies uncountable
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The proof is somewhat roundabout. We do it in two steps:
- Suppose the connected regular space is at most countable. Then, it is a Lindelof space: every open cover has a countable subcover. But any regular Lindelof space is in fact a paracompact Hausdorff space, and hence normal Hausdorff.
- Any connected normal Hausdorff space with at least two points, is uncountable. This follows from the more general fact that any connected Urysohn space with at least two points is uncountable (in fact, its cardinality is at least that of the continuum. For full proof, refer: Connected Urysohn implies uncountable