A denumerable set, S, is an infinite set such that the set of counting numbers can be mapped, one-to-one, with the elements of S.
The set of all positive even numbers, for example, is denumerable.
The mapping x: -> 2x (x >0) is the relevant 1-to-1 mapping.
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{1,2,3,4,5....}
A_{x\x are set of days in a week}
describing of one object
I am a kid, so I might not nail it, but some examples are the dragon curve and julia set.
The set of positive numbers, the set of negative numbers are two examples. Any subsets of these will also not be closed.