That is, (a) Provide an example where f:X→Y is a continuous, bijective map and X is compact, but f is not a homeomorphism.
(b) Provide an example where f:X→Y is a continuous, bijective map and Y is Hausdorff, but f is not a homeomorphism.
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.General observations - why compactness and Hausdorff are required for the theorem to hold
1) Compactness assures that any point has a limit – this impacts the continuity of the inverse of the function f.
2) Hausdorff assures that limit points are unique for sequences (intuitively this means preservation of the topology)....
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