By Czes Kosniowski

This self-contained creation to algebraic topology is acceptable for a couple of topology classes. It includes approximately one area 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a easily grasped subject which supplies a good suggestion of what algebraic topology is). The e-book has emerged from classes given on the collage of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree so one can let the reader to take advantage of it for self-study in addition to a path e-book. The method is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will turn out beneficial as a instructing reduction. This account should be welcomed by way of complex scholars of natural arithmetic at faculties and universities.

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6(c)) we need four dimensions. 6(d). 5 (a) (d) (b) (c) (e) Quotient topology (and groups acting on spaces) 33 not really present, it appears because we live in a three-dimensional world. 7(a),(b)) that a Klein bottle is really just two Môbius strips joined along their common boundary. 7(c),(d). For further intuitive notions recall that the real projective plane RP2 is defined as S2 /- where x = ±x'. x —. x' In this case the northern hemisphere is identified with the southern hemisphere and so we may restrict our attention to the northern hemisphere which is homeomorphic to the disc D2 = { (x,y) E R 2; x2 + y2 <1 } via (x,y,z)-+(x,y) for (x, y, z)E S2 with z 0.

Thus all metrizable spaces are Hausdorff, in particular with the usual topology and any space with the discrete topology is Hausdorff. A space with the concrete topology is not Hausdorff if it has at least two points. 2 ExercIses (a) Let X be a space with the finite complement topology. Prove that X is Hausdorff if and only if X is finite. (b) Let F be the topology on R defined by U Fif and only if for each s U there is a t > s such that [s,t) c U. Prove that (R,J) is Hausdorff. (c) Suppose that X and Y are homeomorphic topological spaces.

By a G-action we shall always mean a left G-action. 7 (a) Suppose that X is a right G-set. For x E X and g E G define gx = x(g'1) Show that this defines a (left) action of G on X. Why does the definition gx = xg fail? (b) Let H be a subgroup of a group G. For hE H, g E G define hg to be hg. Show that this defines an action of H on G. (c) Let G be a group and let,Y(G) denote the set of subsets of G. Show that { ghhEU} ,gEG,UE5°(G) defines an action of G on5'°(G). (d) Let G act on X and define the stabilizer of x E X to be the set {gEG;gx=x}.