Chapter 2: Functions and Relations

In the study of point-set topology, mappings between sets (functions) and the internal structures of sets (relations) serve as the fundamental language. This chapter strictly defines functions, equivalence relations, and order relations. These tools will be employed in later chapters to construct topological spaces, quotient spaces, and order topologies.

1. Functions and Their Basic Properties

In set theory, a function (mapping) is viewed as a specific subset of a Cartesian product characterized by the property of uniqueness.

Definition 2.1 (Function/Mapping)

Let \(A\) and \(B\) be two sets. A function \(f\) from \(A\) to \(B\) is a subset \(S_f\) of the Cartesian product \(A \times B\) such that: for every \(a \in A\), there exists a unique \(b \in B\) such that \((a, b) \in S_f\).

Notation: If \((a, b) \in S_f\), it is denoted as \(b = f(a)\). The mapping itself is denoted as \(f: A \to B\).
Domain: Denoted as \(D(f) = A\).
Image: Denoted as \(I(f) = \{f(a) \mid a \in A\} \subseteq B\).

1.1 Classification of Mappings

Mappings are classified into the following three categories based on how they cover and distinguish elements:

Injective (One-to-one): If for any \(a, a' \in A\), \(a \neq a'\) implies \(f(a) \neq f(a')\).
Surjective (Onto): If \(f(A) = B\), i.e., for every \(b \in B\), there exists at least one \(a \in A\) such that \(f(a) = b\).
Bijective: A mapping that is both injective and surjective.

1.2 Composition and Identity Mappings

For bijections from a set \(A\) to itself, we have the following properties:

Identity Mapping: Defined as \(id_A: A \to A\) where \(id_A(a) = a\).
Composition: If \(f: A \to B\) and \(g: B \to C\), then the composite mapping \(g \circ f: A \to C\) is defined as:

\[ (g \circ f)(a) = g(f(a)) \]

2. Relations and Equivalence Classes

Binary relations describe the connections between elements within a set, where equivalence relations are the core for constructing quotient spaces.

Definition 2.2 (Binary Relation)

A binary relation \(R\) on a set \(A\) is a subset of the Cartesian product \(A \times A\). If \((x, y) \in R\), we write \(xRy\) or \(x \sim y\).

2.1 Equivalence Relations

A relation \(R\) is called an equivalence relation if it satisfies the following three properties:

Reflexivity (Reflexive): For all \(x \in A\), \((x, x) \in R\).
Symmetry (Symmetric): If \((x, y) \in R\), then \((y, x) \in R\).
Transitivity (Transitive): If \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\).

2.2 Equivalence Classes and Partitions

Equivalence Class: The set of all elements that have an equivalence relation \(\sim\) with \(a\), denoted by:

\[ [a] = \{ a' \in A \mid a' \sim a \} \]

Partition Property: Any two equivalence classes are either equal or disjoint. The union of all equivalence classes equals the entire space \(A\).

\[ A = \bigcup_{a \in A} [a] \]

3. Order Relations

Order relations introduce directionality to a set and form the basis for defining the order topology.

Definition 2.3 (Total Order / Linear Order)

A total order relation \(L\) on a set \(A\) is a subset \(L \subseteq A \times A\) that satisfies the following conditions:

(1) Reflexivity: \((x, x) \in L\).
(2) Transitivity: If \((x, y) \in L\) and \((y, z) \in L\), then \((x, z) \in L\).
(3) Comparability (Trichotomy): For any \(x, y \in A\), either \((x, y) \in L\) or \((y, x) \in L\) must hold.

3.1 Lexicographic Order

Let \((A, <_A)\) and \((B, <_B)\) be totally ordered sets. One can define the lexicographic order \(<\) on \(A \times B\): \((a, b) < (a', b')\) if and only if:

\[ a <_A a' \quad \text{or} \quad (a = a' \text{ and } b <_B b') \]

4. Mathematical Induction and Finite Sets

4.1 Mathematical Induction

Let \(P(x)\) be a property concerning natural numbers \(x \in \mathbb{N}_{\ge a}\). If the following hold:

(1) \(P(a)\) is true;
(2) Assuming \(P(y)\) is true for all \(a \le y \le n\) implies that \(P(n+1)\) is true;

Then \(P(x)\) is true for all \(x \ge a\).

4.2 Finite Sets

Definition: A set \(A\) is called a finite set if \(A = \emptyset\), or if there exists a bijection \(f: A \to \{1, 2, \dots, n\}\) for some \(n \in \mathbb{N}\).
Cardinality: If such an \(n\) exists, the size (cardinality) of set \(A\) is said to be \(n\).

5. Preimage and Foundations of Continuity (Tool Supplement)

In topology, we are more concerned with the "pullback" properties of subsets.

Definition 2.4 (Preimage)

For a mapping \(f: X \to Y\) and a subset \(V\) of \(Y\), the preimage of \(V\) is defined as:

\[ f^{-1}(V) = \{ x \in X \mid f(x) \in V \} \]

The preimage operation preserves set operations well:

Preservation of Union: \(f^{-1}(\bigcup V_\alpha) = \bigcup f^{-1}(V_\alpha)\)
Preservation of Intersection: \(f^{-1}(\bigcap V_\alpha) = \bigcap f^{-1}(V_\alpha)\)

This explains why topological continuity is typically defined by "the preimage of an open set is still an open set."