By John B. Conway
This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its mild velocity might be priceless to scholars who're nonetheless studying to put in writing proofs. necessities comprise calculus and at the very least one semester of study, the place the coed has been safely uncovered to the guidelines of simple set idea corresponding to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the genuine line. Appendices are incorporated to bridge the space among this new fabric and fabric present in an research path. Metric areas are one of many extra typical topological areas utilized in different components and are consequently brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the procedure of the booklet to begin with the actual and paintings towards the extra normal. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of concept, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued services, culminating in a improvement of paracompact areas.
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Extra info for A Course in Point Set Topology (Undergraduate Texts in Mathematics)
54 1 Riemann Surfaces Fig. 20 A good pairs of pants decomposition Since every connected stable surface of signature (g, m, n) can be assembled from 2g − 2 + m + n pairs of pants, its area equals areah (S) = 2π(2g − 2 + m + n). 53 is an important milestone towards a compactness theorem for stable Riemannian surfaces. e. it extends over the punctures. Then we can associate to j a unique hyperbolic metric h with finite area. We now know that S can be decomposed isometrically into 2g − 2 + m + n pairs of pants.
22: If h1 , h2 were two hyperbolic metrics on S inducing the complex structure j then we would get two hyperbolic metrics hd1 , hd2 on the double inducing the same complex structure j d . 22 then asserts that the two metrics on S d are identical, therefore h1 ≡ h2 . Assume now that j is a complex structure on S which is of finite type. Recalling the definition, this means that there is a finite extension i : S → Σ such that i∗ j extends to a complex structure jΣ on Σ . Let p ∈ Σ\S. A suitable neighborhood U ⊂ Σ of p is conformally equivalent to the disk D ⊂ C with the standard complex structure i.
If p : S ∪ S ′ → S d is the projection onto the quotient then we get a chart ψα : p −1 (Uα ) → Br (i) ⊂ H + as follows: ψα p(x) := ϕα (x) ϕα′ (x) if x ∈ Uα if x ∈ Uα′ . With f = ϕα ◦ ϕβ−1 : Br+ (i) → Br+ (i) the transition map between coordinate charts ψα , ψβ is of the form (s, t) → f (s, t) −f (−s, t) if s ≥ 0 if s < 0. In general, this is not a smooth map, but it is if f is holomorphic. In our case, f is a local isometry. If we choose an orientation σ on S, the opposite orientation on S ′ so that f is orientation preserving then it is also holomorphic (see below exercise).