Show that, if X is a connected topological space and is continuous, then the image f(X) is an n interval.
Show that, if is a continuous map, then if given a,b,c in with a < b and c between f(a) and f(b), there exists at least one with a and f(x)=c
Let be a continuous map. Show that there exists a point in the circle such that f(x) =f(-x), where is the antipodal point of x. (hint: consider the function defined by g(x)=f(x)-f(-x).)
Please see the attached file for the fully formatted problems.
Connectedness, Continuity, Image, Antipodal Point and Borsuk-Ulam Theorem are investigated.