Talk:Covering map
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User:Charles Matthews There is now some overlap between the content here and at local homeomorphism.
The way covering map has been defined allows it not to be surjective (the condition holds vacuously for points with empty pre-image); the usual definition has a covering map being surjective. I think surjective should be added to the definition since that's what is needed for most purposes.
- Just noticed that one property that a covering map is supposed to have, according to whoever made the page, is being surjective. So I'll add 'surjective' to the definition.
I removed this paragraph of mine:
- The composition of two covering maps need not be a covering map: consider the unit circle S1 as a subset of the complex plane, and for any natural number n define pn : S1 → S1 by pn(z) = z−n. Consider the map p : S1 × N → S1 × N by p(z,n) = (pn(z),n). If N is equipped with the discrete topology and S1 × N carries the product topology, then p is a covering map. The natural projection q : S1 × N → S1 defined by q(z,n) = z is obviously a covering map. The composition qp : S1 × N → S1 is not: no matter how small an open set U you pick in S1, there will always be an n large enough so that pn−1(U) = S1 which cannot be isomorphic to U.
The last statement, pn−1(U) = S1, is false, and that kills the whole argument. I don't know if the composition of two covering maps is always again a covering map. AxelBoldt 14:52, 23 Nov 2003 (UTC)
- Huh? I'm often muddle-headed and confused, but ... the last statement is perfectly true. What's false is the statement that p is a covering map. The problem being that p restricted to to the inverse image S1 does not produce a homeomorphism to U. That is, one can always find an n large enough so that pn(S1) is not equal to U; thus p was never a covering to begin with. Changing -n to +n in the definition would make p into a covering. Interesting example, though. Non-trival fundamental group. linas 15:47, 3 Apr 2005 (UTC)
Would someone who knows what is meant by the "opposite" of a group like to make a stub/redirection? I can't find anything on this.
Special case of dual (category theory); anyway like defining g*h = hg.
Charles Matthews 09:06, 26 Feb 2004 (UTC)
Ah yes, of course. I've added links - that ok?