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 ...continues
Prove that every countable metric space (not empty and not singleton) is disconnected.
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 ?
Prove sqrt(x+1) - sqrt(x) goes to 0 as x goes to infinity
Please see attached file.
See attached file.
See attached pdf for detailed question.
Real Functions of Real Variable, Continuity
Let f:R->R satisfy |f(t)-f(x)|<=|t-x|^2 for any t,x. Prove that (f) is constant.
There are a couple of concepts I need clarification for: 1) If a set has no interior points, then is it necessarily closed? Isn't the empty set considered open? 2) If the graph of f: R -> R is connected, does it have to be continuous? In just Real -> Real aren't these definitions equivalent? Thanks.
