please do the problem 1. True or false For each problem, indicate whether the st

Question

please do the problem 1. True or false For each problem, indicate whether the statement is true or false. Briefly explain your answer. (Formal proof is not required.) If X is a metric space, r>0, and x X, then B_r(x)B_r(x) is not open. If X is a metric space and x, y X, then there is a midpoint m X with d(x, y) = d(x, m) + d(m, y). If X is a metric space, then closed bounded sets are compact. If X is a topological space and x, y X then there is an open set containing x but not y. Complete three of the following four problems. Answers should be proofs given in complete sentences, thoroughly explained. State and prove the General Heine-Boral theorem, and give an example showing the hypothesis of completeness is necessary. A map f: X rightarrow Y of topological spaces is called an open map if for every open se U X, the image f(U) Y is also open. Prove or disprove: If f is open then it is continuous. If f is a homeomorphism then it is open. If f is an open, continuous bijection, it is homeomorphism. If f: R rightarrow R is a continuous surjection, it is open. If f: R rightarrow R is a continuous open surjection, it is a homeomorphism. Let X be a metric space. State the definition of an open subset of X. Define any terms that you

Explanation / Answer

It is given that r is greater than zero, it mean r will always be positive.

That's why its TRUE.

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