Let . What is the maximum order can have?
Every infinite set admits a metric with no isolated points.
Let be a fixed positive integer and let be distinct positive integers such that
Find the maximum possible value of .
Show that the set of numbers of the form is dense in the closed unit interval.
Let be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of in which all matrices have at least one real eigenvalue.
Let be a compact metric space, a continuous function and . Prove that is compact iff is finite.