By Evans L., Thompson R.
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Additional info for Introduction to algebraic topology
Sample text
In the diagram below, the points on opposite edges are equivalent in pairs in the indicated directions. As mentioned earlier, the resulting quotient space is homeomorphic to the so-called Klein 33 34 3. QUOTIENT SPACES AND COVERING SPACES bottle. ) An equivalence relation may be specified by giving a partition of the set into pairwise disjoint sets, which are supposed to be the equivalence classes of the relation. One way to do this is to give an onto map f : X → Y and take as equivalence classes the sets f −1 (y) for y ∈ Y .
THE QUOTIENT TOPOLOGY 37 is Hausdorff. ) Here is another simpler description. ) Consider the inclusion i : S n → Rn+1 − {0} and follow this by the projection to RP n . This map is clearly onto. Since S n is compact, and RP n is Hausdorff, it is a quotient map by the proposition above. (It is also a closed map by the proof of the proposition. ) Since S n is compact, it follows that RP n is compact and Hausdorff. Here is an even simpler description. ) Repeat the same reasoning as above to obtain a quotient map of X onto RP n .
Note that the set p−1 (x) (called the fiber at x) is ˜ necessarily a discrete subspace of X. 17. Let X = S 1 , X 1 n ˜ = Rn . map. More generally, let X = (S ) be an n-torus and let X The map E n is a covering map. 18. Let X = S 1 (imbedded in C), and let X n 1 S . Let p(z) = z . This provides an n-fold covering of S by itself. 19. Let X = C, X n p(z) = z . This is not a covering map. Can you see why? What happens if you delete {0}? ˜ = S n and X = RP n . 20. Let X map discussed earlier. This provides a two sheeted covering.