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| ``Re: Why is inclusion map a topological embedding?''
by AxelBoldt on 2003-11-23 11:30:27 |
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| I assume that you define "topological embedding" as a map that yields a homeomorphism between its domain and its image (where the image carries the subspace topology). Then you're wrong and Mr. Lee is right: every inclusion map is a topological embedding.
In your example, if A is not open in X, then indeed A is open in the subspace topology of A, but A is also open in the (subspace topology) of the image of the inclusion, so there's no problem.
Your example has another flaw: you seem to think that "closed" is the opposite of "open" and that closed sets can never be open. That's not true.
If with "topological embedding" you mean an injective continuous and open map, then you are correct: not all inclusion maps are topological embeddings in this sense. The inclusion map of A into X is a topological embedding in this sense if and only if A is open.
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