5.3.1 Torus If x lies on an edge, then each of the two points in [x] has a half-disc-like neighbourhood. When we identify edges, these neighbourhoods fit together to form disc-like neighbourhoods on the torus, as Figure 105 shows. 5.3 Neighbourhoods 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 creates corresponding disc-like or half-disc-like neighbourhoods of each point y = f(x) of Y. If x is in the interior of X, there is no difficulty: the point x has a disc-like neighbourhood U 5.2.1 Proof We check that Tf satisfies conditions (T1)–(T3) for a topology.
Since (T1)–(T3) are satisfied, Tf is a topology on I(X). Thus (I(X),Tf) is a topological space. We give the topology Tf a sp 5.2 The identification topology 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 can be given the structure of a topological space, and that this space is Hausdorff. Furthermore, we must show that every point has 5.1 Identifying edges of a polygon 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 pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re 4.6.1 Remarks We already know that the characteristic numbers are topological invariants, that is, any two homeomorphic surfaces have the same values for the characteristic numbers. Thus it is solely the converse, namely if two surfaces have the same values for the characteristic numbers then they are homeomorphic, that we have to prove. It follows from the Author(s): 4.6 The Classification Theorem 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 is a topological invariant – it is preserved under homeomorphisms. Let us remind ourselves of these three numbers. A surface may or may not have a boundary, and, if it does, then the boundary has finitely many disjoint pieces. The nu 4.5.2 n-fold toruses We can use a similar technique to find the Euler characteristic of a 2-fold torus. If we cut the surface into two, as shown in Figure 95, and separate the pieces, we obtain two copies of a 1-fold torus with 1 hole, each with Euler characteristic −1. 4.5.1 Surfaces with holes Using this result, we can obtain the Euler characteristic of a surface with any number of holes by successively inserting the holes one at a time. For example, since a closed disc has Euler characteristic 1, it follows that a closed disc with 1 hole has Euler characteristic 0, a disc with 2 holes has Euler characteristic −1, and so on. 4.5 Some general results 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. The Euler characteristic of a surface with an open disc removed is one le 4.4 Historical note on the Euler characteristic 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 more on intuition than would be acceptable today. Attention switched to objects called ‘3-manifolds’, topological s 4.3 The Euler characteristic Subdivisions of surfaces lead to the third number used to classify surfaces, the Euler characteristic. The Euler characteristic χ of a subdivision of a surface is 4.2 Subdivisions 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, edges and faces. It leads to the idea of the Euler characteristic of the surface. All surfaces obtained from polygons by identifying edges arise from a net (of sorts) consisting of a single polygonal face, together with the edges and vertices that remain aft 4.1 Nets on surfaces 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 the Euler characteristic, we need the idea of a subdivision of a surface, which we introduce by first c 3.3 The projective plane 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 by identifying each pair of opposite edges in opposite directions, as shown in 3.2.1 Remarks By ‘contains’, we mean that we can find part of the surface that is homeomorphic to a Möbius band. The edge of the Möbius band does not need to correspond to an edge at the surface, so that a surface without boundary can be non-orientable (as we shall shortly see). When seeking Möbius bands in a surface, it can be helpful to look at all possible closed curves on the surface and thicken these into bands. Remember, fro 3.2 Orientability 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. We can distinguish between a cylinder and a Möbius band by noticing that every cylinder has an ‘inside’ and an ‘outside’, as shown in Author(s): 3.1.1 Inserting half-twists We can insert half-twists into a paper surface whenever a piece of the surface is homeomorphic to a rectangle ABCD with the following properties: the edges AB and CD of the rectangle map to distinct parts of the boundary of the surface, and the edges BC and DA of the rectangle map to non-boundary points of the surface. As illustrated in Author(s): 3.1 Surfaces with twists 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 that resemble a Möbius band. A cylinder or a Möbius band can be formed by gluing together the ends of a rectangular strip or band of paper either with or without twisting the paper before gluing. Does adding further twists to the band before gluing provide any more examples of surfaces Design
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Theorem 10: Disc Removal Theorem
Definition
This unit looks at the process of design – from assessing the complexity of design as an activity, to exposing the difficulty in making general conclusions about how designers work. You will be able to identify innovation in a wide variety of designed objects and evaluate the impact of this innovation. First published on Wed, 27
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