Connected and normal Hausdorff with at least two points implies cardinality at least that of the continuum

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This article gives the statement, and possibly proof, of a topological space satisfying certain conditions (usually, a combination of separation and connectedness conditions) is uncountable.

Statement

Any connected normal Hausdorff space having at least two points is uncountable. In fact, its cardinality is at least equal to the cardinality of the continuum.

Note that the definition of normal space here includes the T1 assumption. The result does not hold for normal-minus-Hausdorff spaces.

Facts used

  1. Urysohn's lemma, which, along with the T1 assumption, tells us that normal Hausdorff spaces are Urysohn spaces.
  2. Connected and Urysohn with at least two points implies cardinality at least that of the continuum

Proof

The proof follows directly by combining Facts (1) and (2).