Algebraic Topology, Aarhus 1978 by J. L. Dupont, I. H. Madsen

By J. L. Dupont, I. H. Madsen

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This would be a continuous function of [0, 1] to {-I, I}, which is impossible. Thus, there must be a point tx with I(t x ) = 0, so l1(t x) = h(tx). The line Lt x drawn tx units from x on the line Dx bisects A. The line Lt x also divides the region B into two (probably not equal) parts. Do this for each point xES. Define two more functions: 91 (x) = area of the portion of B on the x-side of Lt x g2(X) = area of the portion of B on the far side of Lt x Let g(x) = gl(X) - g2(X). Then 9 is a continuous function from the circle S to R Note that for any point XES, the line Dx is the same line as D- x .

3) The union of any finite collection of closed sets in X is closed. 4. 8. Of course, changing the system of neighborhoods used on a space may change which sets are considered to be open, and thus change all the prop~ erties that we have studied such as connectedness, compactness, and continuity, since these are defined in terms of open sets. 1 Open sets and neighborhoods 41 y+b y y-b x-a x x+a Fig. 2. A neighborhood of (x,y) in ]R2 with the open rectangle (or product) topology We need to decide how to compare this topology with the standard topology of ]R2 studied in Chapter 2.

14. 15. 29 and implies that connectedness is a topological property. 23 we defined a version of compactness called sequential compactness. 16) Definition. Let A be a subset of a topological space X. An open cover of A is a collection (9 of open subsets of X so that A lies in the union of the elements of (9, i. , A~ U0 DEe:> A subcover of (9 is a subcollection (9' ~ (9 so that A lies in the union of the elements of (9'. A finite cover (or subcover) is a cover (9 consisting of finitely many sets.

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