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Please answer e and f only
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- Label each of the following statements as either true or false. 3. Let , , and be mappings from into such that . Then .Label each of the following statements as either true or false. 4. Let , , and be mappings from into such that . Then .Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.
- Label each of the following statements as either true or false. 9. Composition of mappings is an associative operation.Let x be a space with at least three points.2-5. For each of the following properties, give an example consisting of two sub- sets X, Y CR2, both considered as topological spaces with their Euclidean topologies, together with a map f: X → Y that has the indicated property. (a) f is open but neither closed nor continuous. (b) f is closed but neither open nor continuous. (c) f is continuous but neither open nor closed. (d) f is continuous and open but not closed. (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.
- Let X and Y be Banach spaces and F:X → Y be a linear map which is continuous and open. Will F always be closed? Will F be always surjective? Give reasons for your answer.Which of the following spaces are subvector spaces?1. Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if inverse image of every closed set is closed in X. Give an example to substantiate the proof.
- Let X be a topological space and Y be a Hausdor ff space Let f, g be a continues mapping of X into Y. Then the set A = (x E XI f(x) = g(x) is closed.Define what it means for a topological space to be completely regular,Let (¥,r) and (¥ ,4) be two topological spaces. Prove that a function f :(X .r)——(¥ ,A) is continuous if and only if the inverse image under / of every A- closed set is a r- closed set.