4.26. (a) Provide a formula for a ho (b) Provide a formula for a homeomorphism between the intervals (-∞, 0] (a, b), with a < b. and (a, b), with a < b. (c) Given the homeomorphisms in Example 4.12 and the first two parts of this exercise, prove that if 11 and 12 are intervals in collection (iii ) in Example 4.12, then 11 and 12 are topologically equivalent. 4.27. Provide an explicit formula for the stereographic projection function in Exam- ple 4.16. 4.28. Prove each of the following statements, and then use them to show that topolog- ical equivalence is an equivalence relation on the collection of all topological spaces: (a) The function id: X → X, defined by id(x) = x, is a homeomorphism. (b) If f : X → Y is a homeomorphism, then so is f-1: Y → X. (c) If f :X + Y and g :Y → Z are homeomorphisms, then so is the composition go f:X → Z. 4.29. Show that R2 – {0} in the standard topology is homeomorphic to Sl R. 4.30. Find two distinct topologies on R such that the first is strictly finer than the second but the two of them are homeomorphic with each other. 4.31. (a) Show that having nonempty open sets that contain finitely many points is a topological property. (b) Prove that the digital line is not homeomorphic to Z with the finite com- plement topology. 4.32. Show that homeomorphism preserves interior, closure, and boundary as indi- cated in the following implications: (a) If f : X → Y is a homeomorphism, then f (Int(A)) = Int( f(A)) for every A CX. (b) If f : X → Y is a homeomorphism, then f (Cl(A)) = Cl(f(A)) for every АСХ. (c) If f : X → Y is a homeomorphism, then f(a(A)) = a (f(A)) for every A CX (+ 4.33. Let X * Y be partitioned into subsets of the form X x {y} for all y in Y. If we let (X x Y)* denote the collection of sets in the partition, show that (X x Y)* with the resulting quotient topology is homeomorphic to Y. 4.34. Let X, Y, and Z be topological spaces. Prove that the three product spaces (X x Y) * Z, X (Y x Z), and X * Y * Z are homeomorphic to each other.
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