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The integers and the rationals have the same cardinality, but the rationals satisfy the property that:

$$ \forall p,q\in\mathbb{Q},\quad \exists r\in\mathbb{Q}\quad \textrm{s.t.}\quad p<r<q, $$

while the integers don't.

Is there a term for this property?

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Since you tagged the question as foundations, you might want to google the terms dense linear order and dense linear order without endpoins. – Git Gud Jun 9 '14 at 19:13
up vote 8 down vote accepted

Such an order on a set is called a dense order.

The notion of dense in topology is closely related to the one in order theory. See this answer.

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