5. Choose whether each of the following statements are true or false. (b) Let (Mi,di) metric spaces for i = 1,2 and f : M1 → M2 be continuous. Then f(U) is open in M2 for all open subset U in M1. (c) Every homeomorphism is uniformly continuous. (d) Every contraction mapping has a fixed point.
5. Choose whether each of the following statements are true or false.
(b) Let (Mi,di) metric spaces for i = 1,2 and f : M1 → M2 be continuous. Then
f(U) is open in M2 for all open subset U in M1.
(c) Every homeomorphism is uniformly continuous.
(d) Every contraction mapping has a fixed point.
(e) An identity function on any metric space is a contraction.
(f) A surjective isometry is a homeomorphism.
(g) If f,g : R→R are uniformly continuous function on R then the produc
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The following definitions are used to obtain the solution.
- Let and be metric spaces. The function is continuous if the preimage of every open set is open.
- Let be a bijection. If the function f and its inverse function f-1 are continuous, then the function f is said to be a homeomorphism.
- A mapping is called a contraction mapping, if there exists a constant c such that
, for all and .
- Let be a function. If there is at least one point x in X such that , then that point is called a fixed point.
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