A Topological Aperitif
Springer Science & Business Media, Apr 3, 2009 - Mathematics - 152 pages
Topologyhasbeenreferredtoas“rubber-sheetgeometry”.Thenameisapt,for the subject is concerned with properties of an object that would be preserved, no matter how much it is stretched, squashed, or distorted, so long as it is not in any way torn apart or glued together. One’s ?rst reaction might be that such animprecise-soundingsubjectcouldhardlybepartofseriousmathematics,and wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlour games. This reaction could hardly be further from the truth. Topology is one of the most important and broad-ranging disciplines of modern mathematics. It is a subject of great precision and of breadth of development. It has vastly many applications, some of great importance, ranging from particle physics to cosmology, and from hydrodynamics to algebra and number theory. It is also a subject of great beauty and depth. To appreciate something of this, it is not necessary to delve into the more obscure aspects of mathematical formalism. For topology is, at least initially, a very visual subject. Some of its concepts apply to spaces of large numbers of dimensions, and therefore do not easily submit to reasoning that depends upon direct pictorial representation. But even in such cases, important insights can be obtained from the visual - rusal of a simple geometrical con?guration. Although much modern topology depends upon ?nely tuned abstract algebraic machinery of great mathematical sophistication, the underlying ideas are often very simple and can be appre- ated by the examination of properties of elementary-looking drawings.
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ambient isotopic angle function antiprisms chapter closed path closure completes the proof construct continuous mapping corresponding cross caps cut-points cylinder define definition deformation edge points equivalent subsets Euclidean set Euler number Euler’s theorem Example f is continuous faces formula four geometrical give given glued graph Hence hexagons homeomorphic sets idea integer interval 0,1 inverse isomorphic joining Jones polynomial Klein bottle Klein handle knot invariant Let f Möbius band n-point neighbourhood non-orientable not-cut-points number of edges open disc opposite points identified orientable origin pair path-connected plane sets Platonic solids polyhedra pre-image Proof Let real line real projective plane rectangle regular polygons regular polyhedron removed rooted trees sets shown shown in Figure shrunk skein relation spherical polyhedron square stereographic projection Suppose surface symmetry Topological Aperitif topological property topological space torus consisting trefoil knot unit circle unknot vertex vertices whereas the complement winding number