By Czes Kosniowski

This self-contained creation to algebraic topology is acceptable for a couple of topology classes. It involves approximately one sector 'general topology' (without its traditional pathologies) and 3 quarters 'algebraic topology' (centred round the basic crew, a comfortably grasped subject which supplies a good suggestion of what algebraic topology is). The ebook has emerged from classes given on the college of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree on the way to allow the reader to take advantage of it for self-study in addition to a path ebook. The method is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will turn out necessary as a instructing reduction. This account may be welcomed by means of complex scholars of natural arithmetic at schools and universities.

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Show that the inclusion map S X is continuous. e. the least number of open sets) such that the inclusion S -+ X is continuous. (d) X is a topological space, S is a subset and i: S X denotes the inclusion map. The set S is given a topology such that for every space Y and map f: Y f: Y S is continuous S if: Y -÷ X is continuous. Prove that the topology on S is the topology induced by the topology on X. (e) Let Y be a subspace of X and let A be a subset of Y. Denote by the closure of A in Y. Clx(A) the closure of A in X and by Prove that Clx(A).

Hint: Try a non-finite set with the discrete topology. ) 7 Compact In this and the next two chapters we shall look at properties of spaces that are preserved under homeomorphisms. One important consequence of this is that if one space has the property in question and another does not, then the two spaces cannot be homeomorphic. The first property is compactness. 7. 1 A cover of a subset S of a set X is a collection of subsets { j E J) of X such that S c EJ } then is U jEJ If in addition the indexing set J is finite said to be a finite cover.

C) Construct examples to show that if X and Y are topological spaces with G acting on them such that X/G Y/G then X and Y are not necessarily homeomorphic. set. For each x E X the stabilizer acts on G and so the quotient G/GX is defined. Show that GIGX is just the set of left cosets of in G. Show that there is a G-equivariant bijection between Cx the orbit of x and G/GX. In the examples that we gave of groups acting on topological spaces the group in question acted continuously; we have a special name for such a space.