PRELIMINARIES FOR GENERAL TOPOLOGY. Contents

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1 PRELIMINARIES FOR GENERAL TOPOLOGY DAVID G.L. WANG Contents 1. Sets 2 2. Operations on sets 3 3. Maps 5 4. Countability of sets 7 5. Others a mathematician knows 8 6. Remarks 9 Date: April 26, 2018.

2 2 D.G.L. WANG 1. Sets 1) set, collection, family, class; element, member. the empty set:. a singleton: {x}. 2) N: the set of nonnegative integers; Z: the set of integers; Q: the set of rational numbers; R: the set of real numbers; C: the set of complex numbers. 3) list notation v.s. set-builder notation: { 1, 2, 3,...} = {x Z x < 0} = {x Z: x < 0}. 4) x A: the element x belongs to the set A, or A contains x. A X: the set A is a subset of the set X, or X includes A. A X: A X and A X. 5) the power set of a set X: 2 X = {A: A X}. A is a proper subset of X: A 2 X \ {, X}. 6) a criterion of equality for sets: A = B A B & B A.

3 3 2. Operations on sets 1) the difference of the sets X and A: X \ A. the complement of A X in X: the set X \ A, denoted A c if X is known well. taking the complement is an involution. 2) intersection, meet: A B. A and B are disjoint: A B =. 3) union, join: A B. A B: disjoint union. 4) Venn diagrams v.s. Euler diagrams. 5) and satisfy the following laws. (a) commutative laws: (b) associated laws: A B = B A and A B = B A. (A B) C = A (B C) and (A B) C = A (B C). (c) distributive laws: ( ) ( ) A B = B Λ, B Λ (d) de Morgan s laws: ( ) c A = (A B) A c 6) symmetric difference of A and B: and and ( ( ) ( A B Λ ) c A = ) B = A c. A B = (A \ B) (B \ A) = (A B) \ (A B). (a) associativity: (A B) C = A (B C). = The expression A B C, or even A, is meaningful. (b) distributivity: (A B) C = (A C) (B C). (c) In general: (A B) C (A C) (B C)., B Λ (A B). Homework 2.1. Prove or disprove the following formulas. i) (A B) C = (A C) (B C). ii) (A B) C = (A C) (B C).

4 4 D.G.L. WANG Hint. The Venn s diagram helps when we are about to explore a set formula. Due to a mistake in an old version of this note, Homework 2.1 is not considered to evaluate your final score. Mar. 4th, 2018.

5 5 3. Maps 1) map, mapping, function: f : X Y or X f Y. image, f-image of an element. We write b = f(a), or a f b, or f : a b. 2) the identity map: id: x x. an inclusion or canonical injection ι: A X: x x, if A X. A transformation: a map f : X X. 3) f is a surjection, or surjective, or onto, if every element of Y is the image of at least one element of X; f is a injection, or injective, or one-to-one, if every element of Y is the image of at most one element of X; f is a bijection, or bijective, invertible, if f is both surjective and injective. 4) the image of a set A X: f(a) = {f(x): x A}. If or Imf: the image f(x) of the whole set X. the preimage of a set B Y : f 1 (B) = {a X : f(a) B}. Proposition 3.1. Let A, A 1, A 2 X and B, B 1, B 2 Y. Let f : X Y. (a) The misleading of the etymology of images and preimages: f(f 1 (B)) B and f 1 (f(a)) A. (b) The set-value map f 1 works well with,, and the complement operation: f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ), f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ), f 1 (B c ) = ( f 1 (B) ) c. and (c) The map f works not that well but fine in the following sense: f(a 1 A 2 ) = f(a 1 ) f(a 2 ), f(a 1 A 2 ) f(a 1 ) f(a 2 ), f ( A f 1 (B) ) = f(a) B. and 5) the composition of maps f : X Y and g : Y Z is the map g f : X Z defined by x g(f(x)). Proposition 3.2. Let f : X Y, g : Y Z, and h: Z W.

6 6 D.G.L. WANG (a) Compositions preserve injection, surjection, and bijection. (b) Associativity: h (g f) = (h g) f. (c) If the composition X f Y g Z is injective, then so is f. If the composition X f Y g Z is surjective, then so is g. 6) A map g : Y X is inverse to a map f : X Y if g f = id X and f g = id Y. invertible map: a map having an inverse map. Proposition 3.3. Invertible maps satisfy the following properties. (a) If an inverse map exists, then it is unique. (b) A map is invertible it is bijective. 7) A transformation f is idempotent: f f = f.

7 7 4. Countability of sets Definition 4.1. We say that two sets have equal cardinality if a bijection between them. A set is countable if it has a subset having the same cardinality as N. Proposition 4.2. Countability has the following properties. (1) Any subset of a countable set is countable. (2) The image of a countable set under any map is countable. (3) The union of a countable family of countable sets is countable. Example 4.3. The sets Z, N 2, and Q are countable. The set R is not countable.

8 8 D.G.L. WANG 1) supremum, denoted sup. infimum, denoted inf. 2) The Greek alphabet: 5. Others a mathematician knows α, β, γ, δ, ɛ(ε), ζ, η, θ(ϑ), ι, κ, λ, µ, ν, ξ, o, π(ϖ), ρ(ϱ), σ(ς), τ, υ, φ(ϕ), χ, ψ, ω. A, B, Γ(Γ ), ( ), E, Z, H, Θ(Θ), I, K, Λ(Λ), M, N, Ξ, O, Π, P, Σ, T, Υ(Υ ), Φ(Φ), X, Ψ(Ψ), Ω(Ω). 3) Some abbreviations in mathematics: a) cf.: an abbreviation for the Latin word confer (the imperative singular form of conferre ), literally meaning bring together, is used to refer to other material or ideas which may provide similar or different information or arguments. b) e.g.: an abbreviation for the Latin phrase exempli gratia, meaning for the sake of example. c) i.e.: an abbreviation for the Latin phrase id est, which means which is to say, that is, or in other words. It is used to denote an example of something stated previously. d) s.t.: such that; so that; subject to. e) w.l.o.g.: without loss of generality. f) w.r.t.: with respect to.

9 9 6. Remarks When something is called a set, this shows, maybe unintentionally, a lack of interest to whatever organization of the elements of this set. The sign is a variant of the Greek letter ɛ, which corresponds to the first letter of the Latin word element. To make the notation more flexible, the formula x A is also allowed to be written in the form A x. Euclid s famous mathematical treatise is also called Elements. The empty set is everywhere. Inclusion is both reflexive and transitive; belonging is neither reflexive nor transitive. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P. R. China address: kwgl@icloud.com

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