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**For instance, the "length" of an open set will be the sum of the lengths of the countable number of open intervals of which it is composed.**"

I don't understand this last sentence.

1) How can there be a "countable number of open intervals" in an open interval? For example, if we have (0,2), I can choose any two numbers x,y such that 0 < x,y <2 and create an open interval from them, right? So I don't understand what they mean by a "countable number of open intervals".

2) The text is telling us the the "'length' of an open set will be

**the sum of the lengths of the countable number of open intervals of which it is composed**". But how can we know the length of those "countable number of open intervals of which it is composed.

I was wondering if anybody could give me an example in order to clarify what this means...

Thanks in advance