Thread: show that M is uncountable

1. show that M is uncountable

If M is connected and has at least two points, show that M is uncountable?
(Hint: Find a nonconstant, continuous, real-value function on M)

I have no idea how to start this proof, need help on it.

2. Re: show that M is uncountable

Originally Posted by wopashui
If M is connected and has at least two points, show that M is uncountable?
(Hint: Find a nonconstant, continuous, real-value function on M)
How do we know that the space is even uncountable?
That is not stated in the question.

3. Re: show that M is uncountable

hmm' i'm not sure what you meant, this is what is said for the entire question

4. Re: show that M is uncountable

Originally Posted by wopashui
hmm' i'm not sure what you meant, this is what is said for the entire question
If the topological space is countable the the statement is false.

5. Re: show that M is uncountable

Originally Posted by Plato
If the topological space is countable the the statement is false.
but we are not into tapology yet, M is just a metric space in this case

6. Re: show that M is uncountable

Then you should have said that! The set {1, 2} with the "indiscrete topology" (the only open sets are the empty set and entire set) is connected and contains exactly two points. Of course this topology is not metric.

In a metric space, given two points, p and q, there exist a continuous function mapping [0, 1] to the space such that f(0)= p and f(1)= q. Since the interval [0, 1] is uncountable, the image of the function is uncountable and so is any set containing it.

7. Re: show that M is uncountable

Originally Posted by HallsofIvy
Then you should have said that! The set {1, 2} with the "indiscrete topology" (the only open sets are the empty set and entire set) is connected and contains exactly two points. Of course this topology is not metric.

In a metric space, given two points, p and q, there exist a continuous function mapping [0, 1] to the space such that f(0)= p and f(1)= q. Since the interval [0, 1] is uncountable, the image of the function is uncountable and so is any set containing it.
thanks, but i think we need to find what this continuous function is though

8. Re: show that M is uncountable

Originally Posted by wopashui
If M is connected and has at least two points, show that M is uncountable?
(Hint: Find a nonconstant, continuous, real-value function on M)

I have no idea how to start this proof, need help on it.
Let d denote the metric on M, and let the two given points in M be p and q. The function f given by f(x) = d(x,p) is continuous from M to the real numbers. Clearly f(p)=0 and f(q)=d(p,q)>0. The aim is to show that the range of this function contains the whole interval [0,d(p,q)].

In fact, if there exists a number r with 0<r<d(p,q) such that f(x) is never equal to r, then the sets U={x : f(x)<r} and V={x : f(x)>r} are nonempty, open, disjoint, and their union is the whole of M. That contradicts the connectedness of M. Therefore the range of f contains the (uncountable) interval [0,d(p,q)], from which it follows that M is uncountable.