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1.1. Topological Spaces 1. Physical quantities: every acceptable value has a neighorhood of accetable values → open sets → topology → continuity, convergence, …. 2. 3. Space point set with structure (organization between its parts). Issues: definition of neighborhood, distinguishability of points, continuity, differentiablity. 4. Topology of a point set S Family T of subsets (open sets) of S such that a. S & Φ T. U j T b. finite j U j T c. j 5. 6. 7. 8. Discrete topology all subsets of S. Indiscrete topolgy { S, Φ }. Topological space (S,T) point set S with topology T. Euclidean space Rn with the usual topology (open-balls). 9. Finite space: T a, a, b, b, c, d , S, . S a, b, c, d , 10. The neighborhood of p S is any set U such that W U , p W and W is an open set. 11. Euclidean space En as a topological space: Point set p p1 , , pn Distance function d p, q p n i 1 i qi 2 Open set open balls 12. Topology basis B U such that for all open set V T and all points p V , there exists some U B such that p U V . 13. First-countable topological space S: neighhoods N j every point p in S has a countable set of such that U p , there exists at least one N j U . 14. B & B’ define the same topology iff U B and p U , there exists some U B such that p U U and vice versa. 15. Second-countable topological space S: S has at least one countable basis. 16. All 2nd countables are 1st countables, but not necessarily the reverse. 17. Induced topology: Let (S,T) be a topological space and XS. The induced (relative) topology is defined as XT. This gives rise to an induced topological space (X, XT). 18. The upper-half space En R n , R n T , where T is the usual topology of En , R n p p1 , , pn Rn ; pn 0 19. En is 1st-countable but not 2nd-countable. ( Open balls tangent to the hyperplane p n 0 must include the tangent point. These balls are uncountable. ) 20. Let (S,T1) & (S,T2) be 2 topological spaces. T1 is weaker than T2 if every member of T1 belongs to T2. T1 is then coarser than T2 & T2 is finer (stronger) than T1. 21. Topology of Minkowski space is not known. One choice is the Zeeman topology: the finest topology on R4 which induces an E3 topology on space sector & E1 topology on time-axis. It’s not 1st countable. 22. The complement of an open set is closed. 23. S & Φ are both open & closed. 24. A connected space is a topological space in which no proper subset is both open & closed → S can’t be decomposed into 2 disjointed open sets. 25. The set of all subsets of S is the power set P(S) or 2S. 26. Discrete topology: T 2S. All sets are both open & closed. E.g., topology induced on the light cone by the Zeeman topology. Since no mapping fron E1 to a discrete space can be continuous, a photon hops from point to point on the light-cone. 1 d p, q 0 28. A prametric is is a mapping: 27. Discrete metric: : S S R such that if pq pq p, p 0 pS 29. A prametric is sufficient to define a topology. 30. Topological (cartesian) product of (A,T’) & (B,T”) with bases U & V, resp, is (AB, T’T”) with basis UV. 31. The n-torus Tn is the cartesian product of n S1. 32. Convergence: filters. Integration: Borel σ-algebra.