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Chapter 1: Set Theory and Logic Foundations

Before diving into General Topology, we must establish a rigorous foundation in set theory. This chapter starts from the ZFC axiom system, reviews fundamental set operations and the concept of mappings, focuses on the cardinality of sets and the famous Cantor's Theorem, and concludes with a brief introduction to relations and partial orders.

1. Introduction to ZFC Axiom System

Modern mathematics widely adopts ZF (Zermelo-Fraenkel) or ZFC (ZF with the Axiom of Choice) as the axiomatic foundation for set theory. The theory is built upon logical symbols (such as \(\vee, \wedge, \neg, \forall, \exists\)) and relational symbols (such as \(\in, =\)).

Basic Axioms of Set Theory (ZF and ZFC)

The following are the 9 fundamental axioms constructing the ZF system (becoming ZFC with the addition of the 10th):

  • 1. Axiom of Extensionality: Two sets are equal if and only if they contain the same elements.
\[ \forall A \forall B (\forall x (x \in A \leftrightarrow x \in B) \rightarrow A = B) \]
  • 2. Axiom of Empty Set: There exists a set that contains no elements, denoted as \(\emptyset\).
\[ \exists A \forall x (x \notin A) \]

  • 3. Axiom of Pairing: Given any two sets \(A\) and \(B\), there exists a set whose elements are exactly \(A\) and \(B\). That is, we can obtain \(\{A, B\}\).

  • 4. Axiom of Union: For any collection of sets \(A\), there exists a set that contains all elements that are members of some set in \(A\). It is denoted as \(\cup A\). For example, if \(A = \{\{1,2\}, \{3\}\}\), then \(\cup A = \{1,2,3\}\).

  • 5. Axiom Schema of Subset / Separation: Given a set \(A\) and a property (formula) \(\varphi(x)\), one can construct a set containing all elements of \(A\) that satisfy \(\varphi(x)\). That is, we obtain \(\{x \in A \mid \varphi(x)\}\).

  • 6. Axiom of Power Set: For any set \(A\), there exists a set containing all subsets of \(A\). This set is called the power set of \(A\), denoted as \(P(A)\) or \(2^A\).

  • 7. Axiom of Infinity: There exists an inductive set containing the empty set and closed under the successor operation. This ensures the existence of infinite sets.

\[ \exists I (\emptyset \in I \wedge \forall x \in I (x \cup \{x\} \in I)) \]

  • 8. Axiom of Replacement: If the domain of a class function is a set, then its range is also a set. That is, for each element \(x\) in \(A\), if there is a unique \(y\) such that \(\varphi(x, y)\) holds, then all such \(y\) form a set.

  • 9. Axiom of Regularity / Foundation: Every non-empty set \(A\) contains an element that is disjoint from \(A\). That is:

\[ \forall A \neq \emptyset, \exists x \in A \text{ s.t. } x \cap A = \emptyset \]
  • 10. Axiom of Choice (AC): For any collection of non-empty and pairwise disjoint sets, there exists a "choice function" that can pick exactly one element from each set in the collection. The system including this axiom is called ZFC.

2. Basic Set Operations and Cartesian Product

In topology, we frequently deal with the intersection and union of a family of sets (set collections).

Definition 1.1 (Union and Intersection of Families)

Let \(\mathcal{F}\) be an arbitrary family of sets (i.e., its elements are themselves sets). The union and intersection of all elements in \(\mathcal{F}\) are defined as follows:

  • Union:
\[ \cup \mathcal{F} = \{x \mid \exists A \in \mathcal{F} \text{ s.t. } x \in A\} \]

  • If \(\mathcal{F} = \{A_1, A_2, \dots, A_n\}\), then \(\cup \mathcal{F} = A_1 \cup A_2 \cup \dots \cup A_n\).

  • Intersection: For a non-empty family of sets \(\mathcal{F}\):

\[ \cap \mathcal{F} = \{x \mid \forall A \in \mathcal{F}, x \in A\} \]
  • If \(\mathcal{F} = \{A_1, A_2, \dots, A_n\}\), then \(\cap \mathcal{F} = \bigcap_{i=1}^n A_i\).

Definition 1.2 (Ordered Pair and Cartesian Product)

  • Ordered Pair: A sequence consisting of two elements \(a\) and \(b\) is called an ordered pair, denoted as \((a, b)\). Its fundamental property is:
\[ (a, b) = (c, d) \iff a = c \wedge b = d \]
  • Cartesian Product: The Cartesian product of two sets \(A\) and \(B\) is defined as the set of all possible ordered pairs:
\[ A \times B = \{(a, b) \mid a \in A, b \in B\} \]

3. Mappings and Cardinality

Mappings are essential tools for connecting different set structures, while "cardinality" generalizes the concept of the "number of elements" from finite sets.

Definition 1.3 (Mapping)

Given sets \(A\) and \(B\). A Mapping \(f\) is a subset of \(A \times B\) such that: for every \(a \in A\), there exists a unique \(b \in B\) such that \((a, b) \in f\).

  • It is usually denoted as \(f: A \to B\).

  • This unique \(b\) is called the image of \(a\) under \(f\), denoted as \(b = f(a)\).

  • For a subset \(A_0 \subseteq A\), its Image is defined as:

\[ f(A_0) = \{f(a) \mid a \in A_0\} \subseteq B \]
  • For a subset \(U \subseteq B\), its Preimage is defined as:
\[ f^{-1}(U) = \{a \in A \mid f(a) \in U\} \subseteq A \]

Cardinality characterizes the size of a set. If there exists a bijection (a mapping that is both injective and surjective) between sets \(A\) and \(B\), then \(A\) and \(B\) are said to be equinumerous (or to have the same cardinality), denoted as \(|A| = |B|\) or \(A \sim B\).

  • Example: There is a one-to-one correspondence between the set of real numbers \(\mathbb{R}\) and the power set of natural numbers \(P(\mathbb{N})\), i.e., \(|\mathbb{R}| = |P(\mathbb{N})|\).

Regarding cardinality, one of the most famous conclusions in set theory is Cantor's Theorem, which states that the power set of any set is "strictly larger" than the set itself.

Theorem 1.1 (Cantor's Theorem)

For any set \(A\), there exists no surjection from \(A\) to its power set \(P(A)\).

In other words, the cardinality of set \(A\) is strictly less than the cardinality of its power set \(P(A)\), i.e., \(|A| < |P(A)|\).

Proof of Cantor's Theorem (Diagonal Argument)

We proceed by contradiction.

Suppose there exists a surjective mapping \(\phi: A \to P(A)\). This means that for every element in \(P(A)\) (i.e., every subset of \(A\)), at least one preimage can be found in \(A\).

Now, we construct a special set \(B \subseteq A\):

\[ B = \{a \in A \mid a \notin \phi(a)\} \]

Since all elements of \(B\) are taken from \(A\), \(B\) is clearly a subset of \(A\), meaning \(B \in P(A)\).

Because \(\phi\) is surjective, there must exist some element \(a_0 \in A\) such that its image is exactly the set \(B\). That is:

\[ \phi(a_0) = B \]

Next, we discuss whether the element \(a_0\) belongs to set \(B\). There are only two possibilities:

  • Case 1: Assume \(a_0 \in B\)

By the definition of set \(B\), if \(a_0 \in B\), then it must satisfy \(a_0 \notin \phi(a_0)\). However, we know \(\phi(a_0) = B\), which implies \(a_0 \notin B\). This leads to a contradiction (\(a_0 \in B \Rightarrow a_0 \notin B\)).

  • Case 2: Assume \(a_0 \notin B\)

Since \(a_0 \notin B\) and \(B = \phi(a_0)\), this implies \(a_0 \notin \phi(a_0)\). However, according to the construction of set \(B\), all elements \(x\) that satisfy \(x \notin \phi(x)\) must be contained in \(B\). Therefore, we must have \(a_0 \in B\). This also leads to a contradiction (\(a_0 \notin B \Rightarrow a_0 \in B\)).

In summary, both cases lead to an irreconcilable logical contradiction. Thus, the initial assumptionβ€”"there exists a surjection from \(A\) to \(P(A)\)"β€”is false. The theorem is proved. \(\square\)

4. Relations and Partially Ordered Sets

In addition to mappings, binary relations on sets are the cornerstones for building complex mathematical structures (such as partial orders and equivalence relations).

Definition 1.4 (Binary Relation and Partial Order)

  • Relation: Let \(X\) be a set. A binary relation \(R\) on \(X\) is a subset of the Cartesian product \(X \times X\), i.e., \(R \subseteq X \times X\). If \((a, b) \in R\), we typically write \(aRb\).

  • Partial Order: Let \(\le\) be a binary relation on \(X\). If \(\le\) satisfies the following three properties, it is called a partial order on \(X\):

  • (1) Reflexivity: For any \(a \in X\), \(a \le a\).

  • (2) Antisymmetry: For any \(a, b \in X\), if \(a \le b\) and \(b \le a\), then \(a = b\).

  • (3) Transitivity: For any \(a, b, c \in X\), if \(a \le b\) and \(b \le c\), then \(a \le c\).

A set \((X, \le)\) endowed with a partial order relation is called a Partially Ordered Set (POSET).

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