algebraic topology a computational approach - kaczynski , mischaikow , mrozek

[...]... that c 62 A0 Again use the fact that c is the least upper bound for A0 and that A0 is open Finally, observe that c 2 I , but c 62 A0 B0 and therefore, that I 6= A0 B0 1.13 Let A and B be connected sets Assume that A \ B 6= Prove that A B is connected 1.14 Show that S 1 is connected 1.15 We say that a topological space X has the xed point property if every continuous map f : X ! X has a xed point,... book) provides an excellent geometric way to proceed from linear algebra to more abstract algebraic structures As was indicated earlier, we do assume that you are familiar with the most basic ideas from linear algebra We shall review them, but as in the previous section we shall present these ideas in a fairly general framework If the words feel unfamiliar don't worry they will be repeated many times throughout... or a > b Assume without loss of generality that a < b (a) Show that the interval a b] I Let A0 := A \ a b] and B0 := B \ a b] (b) Show that A0 and B0 are open in a b] under the subspace topology Let c be the least upper bound for A0 , i.e c := inf fx 2 R j x > y for all y 2 A0 g: (c) Show that c 2 a b] (d) Show c 62 B0 Use the fact that c is the least upper bound for A0 and that B0 is open 24 CHAPTER... is closed, open or neither) Hint: Use an argument similar to that in Example 1.41 1.17 A simple closed curve in Rn is an image of an interval a b] under a continuous map : a b] ! Rn (called a path) such that (s) = (t) for any s < t s t 2 a b] if and only if s = a and t = b Prove that any simple closed curve is homeomorphic to a unit circle 1.2 Linear Algebra Homology theory (what we will learn in this... interior of a set A is the union of all open sets contained in A The interior of A is denoted by int (A) Since the arbitrary union of open sets is open, int (A) is an open set One of the advantages of the abstract de nition of a topology is that it does not explicitly involve a particular norm or distance In fact, there are other norms that can be put on Rn which give rise to the same topology For... (y)jj < Thus, using Theorem 1.31 we can easily show that a variety of simple topological spaces are homeomorphic Proposition 1.33 The following topological spaces are homeomorphic: (i) R, (ii) (a 1) for any a 2 R, (iii) (;1 a) for any a 2 R, (iv) (a b) for any ;1 < a < b < 1 Proof: We begin by proving that R and (a 1) are homeomorphic Let f : R ! (a 1) be de ned by f (x) = a + ex: This is clearly continuous... R2 1.2 LINEAR ALGEBRA 33 1.2.4 Quotient Spaces As will become clear in the next chapter, the notion of a quotient space is absolutely fundamental in algebraic topology We will return to this type of construction over and over again Consider V and W , vector spaces over a eld F , with W a subspace of V Let us set v u if and only if v ; u 2 W: Proposition 1.61 de nes an equivalence relation on elements... set is open As an example consider (0 1] R 1 2 (0 1 ], but given any > 0, B2(1 ) 6 (0 1] Therefore, (0 1] is not open in the standard topology on R The same argument shows that any interval of the form (a b ], a b) or a b] is not open in the standard topology on R Since open sets play such an important role in topology it is useful to be able to refer to the largest open set contained by a set De nition... clearly associative and commutative The zero vector is given by 0i + 0j + 0k: and ;v is given by ; i + (; )j + (; )k Similarly, properties 1-4 of scalar multiplication also hold Nevertheless, since Z is not a eld, Z3 is not a vector space The importance of this last statement will become clear in Chapter 3 To make it clear why in the de nition of a vector space we insist that the scalars form a eld we... Proof: Up to this point, the only topological spaces that have been considered are those of Rn for di erent values of n The abstract de nition of a topology only requires that one begin with a set X So consider X Rn Is there a natural way to specify a topology for X in such a way that it matches as closely as possible the topology on Rn? The answer is yes, but we begin with a more general de nition De nition .