New PDF release: A General Topology Workbook

By Iain T. Adamson

This paintings goals to offer easy topology in an unconventional approach. It provides a assessment of the fundamental definitions including routines with out recommendations or proofs of the theorems partly 1, after which offers the recommendations partially 2, permitting the scholar to check solutions with their very own.

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To finish it suffices to remark that h : X → Y, h(x) = f (x) if x ∈ A g −1 (x) if x ∈ g(B) is a bijection. If X and Y are sets we write |X | ≤ |Y | if an injection X → Y exists. We have just proved the following properties: Reflexivity: |X | ≤ |X | for any set X . Anti-symmetry: |X | ≤ |Y | and |Y | ≤ |X | imply |X | = |Y |. Transitivity: |X | ≤ |Y | and |Y | ≤ |Z | imply |X | ≤ |Z |. We’ll write |X | ≥ |Y | to mean |Y | ≤ |X |. e. that for any X , Y either |X | ≤ |Y | or |Y | ≤ |X |. Given sets X, Y we denote by X Y the set of all maps f : Y → X .

Xn ) = (x0 h(x1 , . . , xn ), x1 , . . , xn ), is a homeomorphism. 25 Prove that the lower half-plane X = {(x, y) ∈ R2 | y ≤ 0} is homeomorphic to Y = X ∪([−1, 1]×[0, 1]). 26 (♥) Check that {z ∈ C | z = a + ib, b > 0} → {z ∈ C | |z| < 1}, z→ z−i z+i is a homeomorphism and describe the inverse. 20 No sphere S n is homeomorphic to a ball D m , n, m ≥ 1. 21 Two products of spheres S n 1 × · · · × S n k and S m 1 × · · · × S m h are homeomorphic only if h = k and n i = m i ∀ i (up to permuting indices).

Every chain C ⊂ A is bounded: considering the obvious element C = ∪{A | A ∈ C}, the maps p : C → X and q : C → Y are injective and thus C bounds C. By Zorn’s lemma A has a maximal element; let’s show that one of the projections p : A → X and q : A → Y is onto. If this were not true there would exist x ∈ X − p(A) and y ∈ Y − q(A); therefore A ∪ {(x, y)} ∈ A, contradicting maximality. Now, if p : A → X is onto then |X | = |A| ≤ |Y |. Similarly if q : A → Y is onto, |Y | = |A| ≤ |X |. 19 Let X be an ordered set with the property that every non-empty subset has a maximum and a minimum.

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