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If someone could just point me in the right direction for this proof or give me some examples that would be great.

(Sorry if this is sloppy but I’ll try to make it as nice as possible)

The problem I'm having trouble with is.

1 Prove that QxQ c R² (The Cartesian product of the rationals in R²) is disconnected

2 Show it is totally disconnected

So far for 1 I have

Let qЄQ

By the Cartesian product we would get a sequence

(q1, q1), (q2, q2), (q3, q3),…,(qn, qn)

(This is where it gets a bit confusing and sloppy. I don’t know if I can take (q1, q1) and just represent it as q)

The numbers in the sequence can be separated as

{x | x > q}∩U1, {x | x < q}∩U2

Therefore by the definition of connected sets it is disconnected

2.

For Q x Q to be totally disconnected it has at least two points and for all distinct points p1, p2 in S, the set S can be separated by two disjoint open sets U1 and U2 into two pieces S∩U1 S∩U2 containing p1 and p2

(q1, q1) Є U1

(q2, q2) Є U2

(q1, q1), (q2, q2) Є S

Therefore they are totally disconnected.