Translation and rotation

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epsilon balls

(See attached file for full problem description with equations) --- • Describe the -balls centered at an arbitrary point for this metric and draw a picture of in he special case of and where x=0 is the origin. • Let be the set of al bounded functions where a function is called bounded if there exists a positive real number K such that for all . Prove that the function defined by setting defines a metric on the set . ---

Homeomorphisms

(See attached file for full problem description with proper equations and symbols) --- 1) Prove that the map GIVEN BY is a homeomorphism between the real line and open interval (-1,1). 2) Let be the map given by a) show that f is a bijection map b) show that f is a continuous map c) If f a homeomorphism? Justify your answer. ---

10 Topology Problems

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connectedness etc

(See attached file for full problem description with symbols and equations) --- • 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).) ---