Topology: An Introduction
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An introduction to topology. Stefan Waldmann is affiliated with Julius Maximilian University of Würzburg, Würzburg, Germany.
family for the patience and support throughout: without this the book would never have been possible. Stefan Waldmann Würzburg June 2014 Symbols M , N , . . . Sets 2 M Power set of M ( M, d ) Metric space with metric d n -Sphere in n -Torus Open ball centred at p with radius r Set of all neighbourhoods of p Sequence of points Topological space with topology Open subset of topological space Closed subset of topological space Subbasis or basis of a topology Subspace
definition of a connected topological space originates form the following observation: Lemma 2.5.1 Consider the closed interval with its usual topology. Suppose we have two open subsets with and . Then necessarily and are just and . Proof Suppose we have two such open subsets in , both non-empty. Without restriction we find and such that . Indeed, the openness of and allows to find more points than just the boundary points and inside and . Now consider all those numbers with and define to
is given, then for all later we have and thus for all . This shows . By symmetry we also get , contradicting . The next proposition shows that we can obtain the closure of a subset by approaching the boundary points by means of net convergence: Proposition 4.1.7 Let be a topological space and let . Then iff there exists a net with converging to . Proof Let . Then for every we find a point . This defines a net converging to as wanted. Conversely, let for and . Then for every there is an
filter than which converges to . Proof Assume that is a cluster point of . Then for all and . We take these sets as basis of a new filter : indeed, this gives a well-defined filter as finite intersections of sets of the form are still non-empty. In fact they are again of this form. Thus taking all with for some and defines a filter , see also Exercise 4.4.8. This filter is finer than both filters and . Thus . Conversely, let with . For every and we have and hence , showing . Hence is a cluster
features of maps: Definition 2.4.5 (Open and closed maps) Let be a map between topological spaces.(i)The map is called open if is open for all open . (ii)The map is called closed if is closed for every closed subset . Suppose that a point yields a closed subset then a constant map is always closed since it simply maps every subset of to a closed subset. It is also continuous but typically not open unless the single point is also an open subset of . Consider the projection onto the first