By Czes Kosniowski
This self-contained creation to algebraic topology is appropriate for a few topology classes. It comprises approximately one area 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the basic team, a simply 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 with the intention to let the reader to take advantage of it for self-study in addition to a direction ebook. The strategy is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will end up beneficial as a educating reduction. This account can be welcomed through complex scholars of natural arithmetic at schools and universities.
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Extra info for A First Course in Algebraic Topology
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). Show that in general Cly(A)*Clx(A).
3 Theorem Let X X Y be the product of two topological spaces. A set W ç such that X X Y is open if and only if for all w W there exist sets is open in X, is open in Y, C W and w X Proof Suppose W is open, then W = U j€J X where J is some indexing set X are open in X,Y respectively. So, if w W then w U, X V, for X is open in X X Y and clearly is some iEJ. Conversely the set U and wEW equal to W. There are obvious projection maps lrx: X X Y -# X and lry: X X Y Y given by (x,y) x and (x,y) y. These are called the product projections.
Induced topology 21 n+ 1 1=1 the topology induced by the usual topology on we may consider the subset S given by R 0. If we give S the induced topology (using the usual topology on R then S is homeomorphic to R". The proof is left as an exercise; alternatively see Chapter 6. It is interesting to look at subspaces of R and try to find which are homeomorphic to each other. For example the intervals [a,bj and [c,d] in R C R3 are homeomorphic. A homeomorphism f is given by f(x) = c + (d— c)(x- a)/(b— a).