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# S24 - 1st December 2004 Munkres 24 Ex 24.2(Morten Poulsen...

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1st December 2004 Munkres § 24 Ex. 24.2 (Morten Poulsen). Let f : S 1 R be a continuous map. Deﬁne g : S 1 R by g ( s ) = f ( s ) - f ( - s ). Clearly g is continuous. Furthermore g ( s ) = f ( s ) - f ( - s ) = - ( f ( - s ) - f ( s )) = - g ( - s ) , i.e. g is an odd map. By the Intermediate Value Theorem there exists s 0 S 1 such that g ( s 0 ) = 0, i.e. f ( s 0 ) = f ( - s 0 ). This result is also known as the Borsuk-Ulam theorem in dimension one. Thus there are no injective continuous maps S 1 R , hence S 1 is not homeomorphic to a subspace of R , which is no surprise. Ex. 24.4. [1, § 17]. Suppose that X is a linearly ordered set that is not a linear continuum. Then there are nonempty, proper, clopen subsets of X : If ( x, y ) = for some points x < y then ( -∞ , x ] = ( , y ) is clopen and 6 = , X . If A X is a nonempty subset bounded from above which has no least upper bound then the set of upper bounds B = T a A [ a, ) = S b B ( b, ) is clopen and 6 = , X . Therefore X is not connected [2, § 23]. Ex. 24.8 (Morten Poulsen). (a) . Theorem 1. The product of an arbitrary collection of path connected spaces is path connected. Proof. Let { A j } j J be a collection of path connected spaces. Let x = ( x j ) j J and y = ( y j ) j J be two points in Q j J A j For each j J there exists a path γ j : [0 , 1] A j between x j and y j , since A j is path connected for all j . Now the map γ : [0 , 1] Q j J A j deﬁned by γ ( t ) = ( γ j ( t )) j J is a path between x and y , hence the product is path connected. ± (b) . This is not true in general: The set S = { x × sin( x - 1 | 0 < x < π - 1 } is path connected, but S = S ( { 0 } × [ - 1 , 1]) is not path connected, c.f. example 24.7.

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S24 - 1st December 2004 Munkres 24 Ex 24.2(Morten Poulsen...

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