Connected Set Topology on R^2 \ Q^2

Let S = R^2 \ Q^2. Points (x,y) in S have at least one irrational coordinate. Is S connected? Can we disprove with a counterexample?

Compact Subset of R^m with Convergent Sequences

Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x. (a) Prove the sequence (x_n) converges. Is this because all the subsequences converge to the same limit? (b) If A is not compact, show that (a) is not necessarily true. If A is not compact, doesn't it imply that (x_n) doesn't necessarily have all subsequences as convergent? Can you help?

Countable Metric Space

Prove that every countable metric space (not empty and not singleton) is disconnected.

Connected Annulus

prove the annulus A={z in (the set)R^2 : r <= |z| <= R} is connected. is it sufficient to show that the annulus is homeomorphic to the circle, and then since circle is connected, so is the annulus ? if so, how do you show it, if not, can you shed light on another method. thank you.

Continuity of a Max Function on [0,1] X [0,1]

Let f(x,y) be a real valued continuous function defined on the unit square [0,1] X [0,1]. Prove g(x)=max{f(x,y) : y in [0,1]} is continuous. --- Can we treat g(x) as a composite function that maps R^2 --> R ?