Algebraic Topology by Mahowald M., Priddy S. (eds.)

By Mahowald M., Priddy S. (eds.)

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Existence of a cyclic transformation transforming linear orders to each other determines an equivalence relation on the set of all linear orders in a set. A cyclic order in a set is an equivalence class of linear orders under the relation of existence of a cyclic transformation. Bx. Prove that for a finite set this definition is equivalent to the definition in the preceding Section. Cx. Prove that the cyclic “counter-clockwise” order on a circle can be defined along the definition of this Section, but cannot be defined as a linear order modulo cyclic transformations of the set for whatever definition of cyclic transformations of circle.

16. Find Cl{1}, Int[0, 1], and Fr(2, +∞) in the arrow. 17. Find Int (0, 1] ∪ {2} , Cl{ n1 | n ∈ N }, and Fr Q in R. 18. Find Cl N, Int(0, 1), and Fr[0, 1] in RT1 . How to find the closure and interior of a set in this space? 19. Does a sphere contain the boundary of the open ball with the same center and radius? 20. Does a sphere contain the boundary of the closed ball with the same center and radius? 21. Find an example in which a sphere is disjoint with the closure of the open ball with the same center and radius.

7. The boundary of a set A equals the intersection of the closure of A and the closure of the complement of A: Fr A = Cl A ∩ Cl(X A). 6′ 6. 8. Let Ω1 and Ω2 be two topological structures in X, and Ω1 ⊂ Ω2 . Let Cli denote the closure with respect to Ωi . Prove that Cl1 A ⊃ Cl2 A for any A ⊂ X. 9. Formulate and prove an analogous statement about interior. 6′ 7. 10. Prove that if A ⊂ B, then Int A ⊂ Int B. 11. Prove that Int Int A = Int A. 12. Do the following equalities hold true that for any sets A and B: Int(A ∩ B) = Int A ∩ Int B, Int(A ∪ B) = Int A ∪ Int B?

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