Is an Uncountable Set and a Continuous Set the Same Thing?

Question

An Uncountable set is a set that has no existence of bijection with $Z$.

Is it the same as a continuous set?

Suppose $[0,1]$ is both uncountable and continuous.

If both are different, please provide an example to clarify it.

Background: I got this doubt because of the following statement from Introduction To Probability by Dimitri P. Bertsekas

Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterize the probability law

Answer 1

No, those are two different concept:

• countable comes from set basic theory; the main property, one might argue, of a set, is that it has cardinality: it has a number of elements. This number can be 0 (if the set is empty), infinite or any number in between. The way we compare the size of two sets is by trying to construct isomorphisms (bijections) between them. As it turns out, it is possible to prove that there is no bijection between $\mathbb{N}$ and $\mathbb{R}$, for example which is why we distinguish between countable and uncountable

• continuous (or in the case of sets: connected) comes from topology, which is set theory plus something more, loosely speaking. The idea is that given a set, $A$, we also have a collection of subsets of $A$, which fulfills certain conditions; we call this set of subsets the topology of $A$, often denoted $\tau_A$. Armed with a topology, we can now define the concepts continuous (for functions) and connected (for sets). It is possible (and sometimes even useful) to define topologies for discrete sets.

There are some quite good articles on wikipedia about set theory and topology, if you want to dig a bit deeper. For set theory, I always recommend P.R. Halmos' Naive Set Theory, which I think is an excellently written book.

Answer 2

No , continuity and uncountabililty are two different concepts. We check continuity of functions while countability/ uncountability is checked for sets.

Answer 3

As has been pointed out in one of the comments, the term continuous set is not generally used in Western literature.

That being said, in the reference you provided, I found on page 4 in regards to the set $[0,1]$: Note that the elements $x$ of the latter set take a continuous range of values, and cannot be written down in a list; such a set is said to be uncountable.

Hence, the author holds up an an example the "continuous set" $[0,1]$ and because its elements cannot be listed out, makes the implication that it is an uncountable set. This is true.

However, in order to go the other way, that is, is every uncountable set continuous?---the author needs to have provided a definition of what he means by a continuous set.

I did not see such a definition.

If he did provide one, please post it so then we can compare if uncountable sets imply continuous sets according to his definition.

In the meantime, consider the set of irrational numbers. Certainly, it is an uncountable set, but is it a continuous set? Again, we need the author's definition for such terms are not universally in use.