Topology Question Involving Compactness, Hausdorff and Homeomorphism

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Question

Let f:X→Y be a continuous, bijective map. Recall the following theorem: If X is compact and Y is Hausdorff, then f is a homeomorphism. Show that both assumptions are necessary for the theorem to hold.
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.

Solution Preview

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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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