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Chapter 4: Open Sets and Topological Structure

In previous chapters, we explored the basic properties of sets, mappings, and metric spaces. Based on this, this chapter will start from open sets in metric spaces, gradually stripping away the concept of distance to establish an axiomatic system for general topological spaces, with a focus on the topological characterization of continuity and the construction of bases for a topology.

1. Open and Closed Sets in Metric Spaces

In metric spaces (or distance spaces), the foundation of all topological properties is the concept of a "ball."

Definition 4.1 (\(\epsilon\)-Ball)

Let \((S, d)\) be a metric space. For any \(x \in S\) and any real number \(\epsilon > 0\), the \(\epsilon\)-ball centered at \(x\) with radius \(\epsilon\) is defined as:

\[ B(x, \epsilon) = \{y \in S \mid d(y, x) < \epsilon\} \]

Based on the \(\epsilon\)-ball, we can strictly define open sets and closed sets:

Definition 4.2 (Open and Closed Sets)

  • Open Set: A subset \(P \subseteq S\) is called an open set if for every point \(x \in P\), there exists a constant \(\epsilon > 0\) such that \(B(x, \epsilon) \subseteq P\).

  • Closed Set: A subset \(C \subseteq S\) is called a closed set if its complement \(S \setminus C\) is an open set.

Basic Examples and Properties:

  • The entire space \(S\) and the empty set \(\emptyset\) are both open and closed.

  • Arbitrary Unions: If every \(A_i\) in the family \(\{A_i\}_{i \in I}\) is an open set, then their union \(\bigcup_{i \in I} A_i\) is also an open set.

  • Finite Intersections: The intersection of finitely many open sets must be an open set.

Proof: Finite Intersections of Open Sets are Open

We only need to prove that the intersection of two open sets is open, and then the conclusion can be extended to any finite number of sets by mathematical induction. Let \(A_1, A_2, \dots, A_k\) be a finite number of open sets.

Consider their intersection \(\bigcap_{i=1}^k A_i\). If this intersection is the empty set \(\emptyset\), the conclusion holds naturally since the empty set is open.

If the intersection is non-empty, take any point \(x \in \bigcap_{i=1}^k A_i\).

This means that for every \(i \in \{1, 2, \dots, k\}\), \(x \in A_i\).

Since each \(A_i\) is an open set, by the definition of open sets, there must exist a corresponding \(\epsilon_i > 0\) such that \(B(x, \epsilon_i) \subseteq A_i\).

We simply take the minimum of these radii:

\[ \epsilon = \min\{\epsilon_1, \epsilon_2, \dots, \epsilon_k\} \]

Since the collection is finite, \(\epsilon\) must be a real number strictly greater than 0. At this point, for all \(i\), we have \(B(x, \epsilon) \subseteq B(x, \epsilon_i) \subseteq A_i\).

Therefore, \(B(x, \epsilon) \subseteq \bigcap_{i=1}^k A_i\). This proves that any point in the intersection contains an \(\epsilon\)-ball that lies entirely within the intersection; thus, a finite intersection is an open set. \(\square\)

2. Topological Characterization of Continuity

In calculus, continuity is usually described in \(\epsilon-\delta\) language. In more general spaces, we can use open sets to characterize continuity in an entirely equivalent way.

Definition 4.3 (Continuity of Mappings)

Let \(f: S \to M\) be a mapping between two spaces.

  • Local Continuity at a Point: A mapping \(f\) is said to be continuous at a point \(x_0 \in S\) if for any open set \(V\) in the codomain \(M\) containing the image point \(f(x_0)\) (i.e., \(f(x_0) \in V\)), there exists an open set \(U\) in the domain \(S\) such that \(x_0 \in U \subseteq f^{-1}(V)\).

  • Global Continuity: A mapping \(f\) is said to be continuous on the entire space \(S\) if and only if it is continuous at every point in \(S\). Equivalently, \(f\) is continuous if and only if for any open set \(V\) in \(M\), its preimage \(f^{-1}(V)\) must be an open set in \(S\).

3. Axiomatic Definition of General Topological Spaces

Once we base the definition of continuity entirely on open sets, the distance \(d\) is no longer necessary. We can directly axiomatize the concept of "open sets" to define general topological spaces.

Definition 4.4 (Topology and Topological Space)

Let \(X\) be a non-empty set. A topology \(\tau\) on \(X\) is a subcollection of the power set \(\mathcal{P}(X)\) (i.e., \(\tau\) contains some subsets of \(X\)) that satisfies the following three axioms:

  • (1) The entire space \(X\) and the empty set \(\emptyset\) belong to \(\tau\). That is, \(\emptyset, X \in \tau\).

  • (2) Closure under Arbitrary Unions: If \(\{A_i\}_{i \in I} \subseteq \tau\), then their union also belongs to \(\tau\). That is, \(\bigcup_{i \in I} A_i \in \tau\).

  • (3) Closure under Finite Intersections: If \(A_1, A_2, \dots, A_k \in \tau\), then their finite intersection also belongs to \(\tau\). That is, \(\bigcap_{i=1}^k A_i \in \tau\).

The set \(X\) endowed with the topology \(\tau\) is called a topological space \((X, \tau)\). The elements in \(\tau\) are called open sets of \(X\) with respect to this topology.

If the complement \(X \setminus S\) of a subset \(S \subseteq X\) is an open set (i.e., belongs to \(\tau\)), then \(S\) is called a closed set.

Example: Let \(X = \{a, b\}\). We can define a topology \(\tau = \{\emptyset, X, \{a\}\}\). In this topological space, \(\{a\}\) is an open set, and its corresponding complement \(\{b\}\) is a closed set.

Closed sets have properties dual to those of open sets:

  • \(\emptyset\) and \(X\) are both closed sets.

  • The intersection of any number of closed sets \(\bigcap_{i \in I} C_i\) is still a closed set.

  • The union of a finite number of closed sets \(\bigcup_{i=1}^k C_i\) is still a closed set.

4. Bases for a Topology

To describe a topology \(\tau\), we often do not need to list all the open sets within it. Instead, we can generate the entire topology through a smaller collection of setsβ€”a basis.

Definition 4.5 (Basis for a Topology)

Let \((X, \tau)\) be a topological space. A collection of open sets \(\mathcal{B} \subseteq \tau\) is called a basis for the topology \(\tau\) if for any open set \(S \in \tau\) and any point \(x \in S\), there exists a basis element \(B \in \mathcal{B}\) such that:

\[ x \in B \subseteq S \]

From this, it follows that every open set \(S\) in the topology \(\tau\) can be expressed as the union of some basis elements, i.e., \(S = \bigcup_{i \in I} B_i\).

Not every collection of sets can serve as a basis for some topology. The following theorem provides the criteria:

Theorem 4.1 (Criterion for a Basis)

Let \(\mathcal{B} \subseteq \mathcal{P}(X)\) be a collection of subsets of \(X\). Then \(\mathcal{B}\) can be a basis for some topology on \(X\) if and only if it satisfies the following two conditions:

  • (1) Coverage: The union of all basis elements must cover the entire space \(X\). That is:
\[ X = \bigcup_{B \in \mathcal{B}} B \]
  • (2) Internal Consistency: If two basis elements \(B_1, B_2 \in \mathcal{B}\) are chosen, and any point \(x \in B_1 \cap B_2\) is taken from their intersection, then there must exist a third basis element \(B_3 \in \mathcal{B}\) such that:
\[ x \in B_3 \subseteq B_1 \cap B_2 \]

If a collection of sets \(\mathcal{B}\) satisfies the two conditions above, then the topology \(\tau\) generated by it contains exactly all subsets formed by arbitrary unions of elements in \(\mathcal{B}\).

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