By Allan J. Sieradski

This article is an creation to topology and homotopy. themes are built-in right into a coherent complete and built slowly so scholars aren't crushed. the 1st 1/2 the textual content treats the topology of entire metric areas, together with their hyperspaces of sequentially compact subspaces. the second one half the textual content develops the homotopy classification. there are various examples and over 900 workouts, representing a variety of trouble. This booklet will be of curiosity to undergraduates and researchers in arithmetic.

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**Example text**

29 Orientable surface of genus 2 with four holes. surface, every edge is an edge of exactly two triangles. However, if a bordered surface is triangulated, some edges will be edges of only one triangle. Such edges will be contained in the boundary. It is a theorem, which we will assume without proof, that every compact bordered surface may be triangulated (for a proof, see Ahlfors and Sario [1], Chapter I, Section 8). Let M be a compact bordered surface, with a given triangulation. 30 Two-Dimensional Manifolds Barycentric subdivision of a triangle.

We can regard a triangulated surface as having been constructed by gluing together the various triangles in a certain way, much as we put together a jigsaw puzzle or build a wall of bricks. Because two different triangles cannot have the same vertices we can specify completely a triangulation of a surface by numbering the vertices, and then listing which triples of vertices are vertices of a triangle. Such a list of triangles completely determines the surface together with the given triangulation up to homeomorphism.

4 (a) The sides of a regular octagon are identiﬁed in pairs in such a way as to obtain a compact surface. Prove that the Euler characteristic of this surface is g —2. (b) Prove that any surface (orientable or nonorientable) of Euler characteristic g -—2 can be obtained by suitably identifying in pairs the sides of a regular octagon. , it is a hexagon) and such that distinct regions have no more than one side in common. 6 Let SI be a surface that is the sum of m tori, m g 1, and let S, be a surface that is the sum of n projective planes, n _2_ 1.