# Homework Help: Please check my Contrapositive statement

1. Feb 14, 2013

### Bachelier

I am trying to write a CP for:

Every connected M. Space with at least 2 points is uncountable.

Restatement:

if a MS X is connected with |X|≥ 2 => X is uncountable.

Contrapositive:

a MS X has only one point => X is not connected.

Thanks

Last edited: Feb 14, 2013
2. Feb 14, 2013

### Bachelier

I guess the correct statement is that:
if X has only one point it is separated.

3. Feb 14, 2013

### Bachelier

How do I prove this though. the part about the singleton

4. Feb 14, 2013

### Dick

If a space has only one point, it's clearly countable isn't it? Forming the CP isn't the challenge, it's showing a metric space with two points in it that's pathwise connected is uncountable. Any ideas on that one?

5. Feb 15, 2013

### micromass

You seemed to have changed "connected" into 'pathwise" connected. The OP was talking about just connected. The statement is true in both cases though (as I'm 100% sure you know), but it's much easier to prove if you take pathwise connected.

6. Feb 15, 2013

### Dick

Yep, true. I actually don't think it's hard for either form of connected. Thanks!

7. Feb 15, 2013

### pasmith

A metric space consisting of a single point is necessarily connected, is it not? The only non-empty open subset is the whole space.

8. Feb 15, 2013

### D H

Staff Emeritus
Given the statement "if a then b" (equivalently, ab), the contrapositive is "if not b then not a" (¬b → ¬a).

Your right hand side in the first statement is "X is uncountable". The negation is "X is countable", which is how your contrapositive statement should start.