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Q1: When a question asking me to show a set consists of countably many point, do it mean infinite countably many or both infinite or finite case.

Q2:Also, to show that a set consist of countably many point, is it enough to show that the set is countable or i have to show somethings more?

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2 Answers

One of the official definitions of countable set is a set $A$ such that there is a bijection between $A$ and a subset of the natural numbers. That definition includes the finite sets.

If a set is countable but not finite, it should be called countably infinite.

But people often sloppily write "countable" when in principle they should write "countably infinite." So if the phrase is in a problem on an assignment, you may have to ask the instructor. However, the answer may be clear from the context. And if, for example, you have shown that a set is in one to one correspondence with the natural numbers, you will have shown that it is countable. For you will have proved the stronger result that the set is countably infinite.

As to the second question, "countable" and having "countably many points" mean the same thing.

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So, what do you think about the second question? is it enough to show a set is a countable set and conclude immediately that it has contably many point of i have to add somethings more? –  ABC Sep 27 '12 at 3:09
    
"Countable" and "countably many points" mean the same hing. –  André Nicolas Sep 27 '12 at 3:11
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For 1: the common use of the word countable includes both finite and countably infinite (the definition is that $A$ is countable if $|A|\leq\aleph_{0}$)

For 2: yes, it is sufficient as you show this by definition

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