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What is countability?

What is countability? Here are some definitions.

Noun
  1. The quality of being countable.
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Examples
The PN space under the strong topology is a Hausdorff space and satisfies the first countability axiom.
At the heart of all fundamental theories is the concept of countability.
Appendices are included on Mathspeak, countability, and decadic notation.
But as a delegate I am very aware of the countability I have to represent members' views, rather than just my own and I need to discuss this widely with others.
Throughout the article X will denotes the topological Hausdroff group, written additively, which satisfies the first axiom of countability.

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