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Time: 20 hours
Level: Advanced
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 Introduction Resource- This unit is concerned with a special class of topological spaces called surfaces. Common examples of surfaces are the sphere and the cylinder; less common, though probably still familiar, are the torus...
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| | 1 Topological spaces and homeomorphism
- 1 Topological spaces and homeomorphism Resource
- Two topological spaces (X, TX) and (Y, TY) are homeomorphic if there is a bijection f : X → Y that is continuous, and whose inverse f−1 is also continuous, with respect to the given topologies; such a...
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- 2.1 Surfaces in space Resource
- In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon...
- 2.2 Surfaces in space Resource
- In this section we present a wide range of examples of surfaces in space.
- 2.3 Paper-and-glue constructions Resource
- In this section we show how to construct surfaces by taking a piece of paper in the shape of a polygon and gluing some of its edges together. The surfaces that we obtain occupy a central position in this...
- 2.4 Homeomorphic surfaces Resource
- As we stated in Section 1, our aim is to classify surfaces up to homeomorphism. So it is worthwhile spending a little time examining what sorts of transformations of surfaces are homeomorphisms. We shall...
- 2.5 Defining surfaces Resource
- In Section 2.1 we gave a provisional definition of a surface. The aim of this section is to formalise that definition. To do that, we need to specify three further requirements of a candidate topological...
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| | 3 The orientability of surfaces
- 3.1 Surfaces with twists Resource
- In Section 3 we study the orientability of surfaces from an informal point of view. In particular, we take a detailed look at the projective plane and its properties. We start by examining some surfaces...
- 3.2 Orientability Resource
- The idea of orientability is another fundamental concept that we need for the study of surfaces. To illustrate the underlying idea, we consider two familiar surfaces – a cylinder and a Möbius band.
- 3.3 The projective plane Resource
- We now consider one of the most important non-orientable surfaces – the projective plane (sometimes called the real projective plane). In Section 2 we introduced it as the surface obtained from a rectangle...
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| | 4 The Euler characteristic
- 4.1 Nets on surfaces Resource
- In Section 4 we introduce the third of the numbers we associate with a surface – the Euler characteristic. This is used in the Classification Theorem, which we state at the end of the section. To define...
- 4.2 Subdivisions Resource
- In this subsection we formalise the idea of a net by introducing a useful concept called a subdivision of a surface. This is a standard kind of net drawn on a surface, and is defined in terms of vertices,...
- 4.3 The Euler characteristic Resource
- Subdivisions of surfaces lead to the third number used to classify surfaces, the Euler characteristic.
- 4.4 Historical note on the Euler characteristic Resource
- A little history is instructive here, because it shows how difficult Theorem 9 really is. By 1900 the classification of compact surfaces was well understood, although proofs of the major theorems relied...
- 4.5 Some general results Resource
- We next establish some general results about Euler characteristics. We start with a theorem that tells us what happens to the Euler characteristic of a surface when we remove an open disc.
- 4.6 The Classification Theorem Resource
- In this subsection we state the Classification Theorem for surfaces, which classifies a surface in terms of its boundary number β, its orientability number ω and its Euler characteristic χ, each of which...
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- 5.1 Identifying edges of a polygon Resource
- In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in...
- 5.2 The identification topology Resource
- Our aim is to show that the object that we produce when we identify some or all the edges of a polygon is a surface. Therefore, by the definition of a surface given in Section 2.5, we must show how it...
- 5.3 Neighbourhoods Resource
- We know that a polygon X is a surface and so each point x in X has a disc-like or half-disc-like neighbourhood. We shall show that a map f that identifies edges of a polygon to create an object Y automatically...
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| | References and Acknowledgements | |
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