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A question from Dugundji's book which I don't even understand the statement.

Describe the relative topology of $\{z: |z|=1\}$ as a subspace of ${\mathbb{R}}^{2}$.

What do they mean by "describe"? I don't understand what they are asking for. Can you please clarify?

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I don't know what describe means either. The open sets are countable unions of disjoint open arcs; that's one way to describe the topology. It's a compact connected 1-dimensional manifold (the only one up to homeomorphism). –  Jonas Meyer Dec 22 '10 at 6:30
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I agree that the exercise is not very precisely stated, and I don't myself know exactly what Dugundji was intending here (and not, I think, due to any lack of understanding of basic topology).

The best I can do is to suggest a reasonable problem which could be what Dugundji had in mind: show that "open intervals" (edit: Jonas Meyer's terminology open arcs is better) on the unit circle form a base for the subspace topology.

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Ah, ok. Thank you, it is clear now. –  Paulo Dec 22 '10 at 6:41
    
@Paulo: you're welcome. If you're fully satisfied with this answer, please click the "accept answer box" to accept it. (This gives me a little "reputation", but more importantly it keeps things tidy on the site. It is very convenient to be able to tell whether a question has been answered or not, and unanswered questions periodically get kicked to the top of the queue.) –  Pete L. Clark Dec 22 '10 at 6:48
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