Understanding Disconnectedness in Countable Metric Spaces

[Bachelier]

We know that every discrete metric space with at least 2 points is totally disconnected.
Yet I read this:
A MS that is countable with more than 2 pts is disconnected. Is it that I'm misreading this statement. It sounds like if it has 2 or less points it is connected?
more means greater than.

 

[micromass]

Well, a metric space with 2 points is disconnected. So the statement should be: a countable metric space with 2 or more points is disconnected.

 

[Bachelier]

what is the equivalent statement of:

no discrete MS with more than 2 pts is connectd?

 

[micromass]

Every discrete space with more than 2 points is disconnected?

 

[Bachelier]

Every discrete space with more than 2 points is disconnected?

Exactly, which is false. because of we have a discrete MS X with 2 pts it is also disconnected. Yet the statement asks every discrete MS with > 2 pts.

 

[micromass]

That doesn't make the statement false! If I say: for all numbers n greater than 2 it holds that n+0=n, then this statement is true. Whether it holds for other numbers n does not change the validity of the statement.

So the truth of the statement "every countable metric space with more than 2 points is disconnected" is independent of what happens for 2 points!

 

[Bachelier]

That doesn't make the statement false! If I say: for all numbers n greater than 2 it holds that n+0=n, then this statement is true. Whether it holds for other numbers n does not change the validity of the statement.

So the truth of the statement "every countable metric space with more than 2 points is disconnected" is independent of what happens for 2 points!

Makes sense. thanks