Uncountable infinite sets that are not continuous

1. Sep 13, 2011

jashua

Can you give some examples of the infinite sets that are uncountable and that are not continuous?

I know the infinite sets that are countable and discrete, and I know the continuous sets, but couldn't find an example for the above situation.

2. Sep 13, 2011

Ray Vickson

Define what you mean by a "continuous set". Probably, various people have different ideas about what that could mean.

RGV

3. Sep 13, 2011

jashua

A set is called continuous if it contains an infinite number of elements equal to the number of points on a line segment.

A set is called discrete if it contains a finite number of elements or an unending sequence of elements with as many elements as there are whole numbers.

4. Sep 13, 2011

Dick

How about the set of all subsets of the points on a line segment?

5. Sep 13, 2011

jashua

I'm thinking about your example :) The number of all the subsets are infinite. There are also infinite number of discrete subsets and infinite number of continuous subsets. And then how do we deduce that the overall set is not continuous ? Is the reason that we have infinite number of discrete subsets?

I'm still confused. In fact, the question is related to the discrete/continuos random variables in probability theory. The question can be restated as: Is there a sample space which is not continuos and consists of uncountable infinite number of samples? Childer's Probability Theory book says yes, and it is beyond the scope of the book. However, it does not give an example.

Last edited: Sep 13, 2011
6. Sep 13, 2011

Tomer

You said:
I guess by "infinite number equal to..." you mean that the two sets are of the same Cardinality.
Dick's example is a set which is of a higher cardinality than the real segment (cause it's the power set of the real segment), and therefore, according to your definition, not continuous.