Chapter 5: Induced Topology and Basic Topological Operations
In topology, we need not only to study given isolated spaces but also to investigate how to naturally construct new topological spaces from known ones. This chapter focuses on the two most fundamental construction methods: subspace topology and product topology. Additionally, we will delve into basic set operations within topological spaces, including the concepts of interior, closure, and boundary, revealing the dual connection between open and closed sets.
1. Subspace Topology (Relative Topology)
When we focus on a subset of a topological space, we can "restrict" the topological structure of the entire space to this subset, thereby endowing it with a natural topology.
Definition 5.1 (Subspace Topology / Relative Topology)
Let \((X, \tau)\) be a topological space, and let \(A \subseteq X\) be a subset of \(X\). We define a new topology \(\tau_A\) on \(A\), whose elements consist of the intersections of \(A\) with the open sets of the entire space:
This topology \(\tau_A\) is called the subspace topology induced by \(X\) on \(A\), and the pair \((A, \tau_A)\) is called a topological subspace of \((X, \tau)\). The elements in \(\tau_A\) are referred to as relatively open sets (or open sets in \(A\)).
Subspace topology possesses the property of transitivity. If \(A' \subseteq A \subseteq X\), we can either induce the topology on \(A\) first and then on \(A'\), or induce it directly from \(X\) to \(A'\). These two methods yield identical results:
(Supplementary Note): Not only can open sets be obtained via intersection, but closed sets in a subspace also possess the same structure. A set \(C_A \subseteq A\) is a closed set in the subspace \(A\) if and only if there exists a closed set \(C\) in the entire space \(X\) such that \(C_A = C \cap A\).
2. Product Topology
In addition to taking subsets, we can construct new spaces by taking the Cartesian product of two topological spaces.
2.1 Basis for Product Topology
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be two topological spaces. Consider their Cartesian product \(X \times Y\). We wish to endow \(X \times Y\) with a natural topology.
Definition 5.2 (Natural Basis for Product Topology)
Define a collection of sets \(\mathcal{B}_{X \times Y}\) on \(X \times Y\), consisting of all "open rectangles":
We can rigorously prove that \(\mathcal{B}_{X \times Y}\) indeed forms a basis for a topology on \(X \times Y\):
-
(1) Coverage: Since the entire spaces \(X \in \tau_X\) and \(Y \in \tau_Y\) are open, \(X \times Y \in \mathcal{B}_{X \times Y}\). The basis elements clearly cover the entire \(X \times Y\).
-
(2) Internal Consistency: Let \(B_1, B_2 \in \mathcal{B}_{X \times Y}\), such that \(B_1 = U_1 \times V_1\) and \(B_2 = U_2 \times V_2\). For their intersection:
\[ B_1 \cap B_2 = (U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2) \]Since topologies are closed under finite intersections, \(U_1 \cap U_2 \in \tau_X\) and \(V_1 \cap V_2 \in \tau_Y\). This means the intersection \(B_1 \cap B_2\) itself belongs to \(\mathcal{B}_{X \times Y}\), naturally satisfying the consistency condition for a basis.
Definition 5.3 (Product Topology)
The topology generated by the basis \(\mathcal{B}_{X \times Y}\) is called the Product Topology on \(X \times Y\), denoted as \(\tau_{X \times Y}\).
2.2 Constructing Product Topology using Bases of Original Spaces
If we do not wish to use all open sets of the original spaces, we can generate the product topology using only their bases.
Lemma 5.1
If \(\mathcal{B}_1\) is a basis for the topological space \(X\) and \(\mathcal{B}_2\) is a basis for the topological space \(Y\), then their product:
constitutes a basis for the product topology \(\tau_{X \times Y}\).
Proof Sketch: For any open set \(W\) in the product space and any point \((x, y) \in W\), by the definition of product topology, there must exist an open set \(U\) in \(X\) and an open set \(V\) in \(Y\) such that \((x, y) \in U \times V \subseteq W\). Since \(\mathcal{B}_1\) and \(\mathcal{B}_2\) are bases for \(X\) and \(Y\) respectively, we can find basis elements such that \(x \in B_1 \subseteq U\) and \(y \in B_2 \subseteq V\). Thus \((x, y) \in B_1 \times B_2 \subseteq U \times V \subseteq W\). The lemma is proven.
2.3 Compatibility between Subspace and Product
These two fundamental construction operations are highly compatible.
Theorem 5.1 (Commutativity)
Let \(A \subseteq X\) and \(D \subseteq Y\). On the Cartesian product \(A \times D\), we have two ways to define a topology:
-
Method 1: Take the subspace topologies \(\tau_A\) and \(\tau_D\) respectively, then form the product topology \(\tau_A \times \tau_D\).
-
Method 2: Form the product space \(X \times Y\) with the product topology \(\tau_{X \times Y}\) first, then take \(A \times D\) as its subspace to induce the relative topology \((\tau_{X \times Y})_{A \times D}\).
These two methods yield exactly the same topology.
(Supplementary Note: Projection Maps): The deep significance of the product topology is that it is the coarsest topology (having the fewest open sets) that makes the natural projection maps \(\pi_X: X \times Y \to X\) (defined by \(\pi_X(x, y) = x\)) and \(\pi_Y: X \times Y \to Y\) continuous.
3. Interior, Closure, and Boundary
In a topological space, we can "approximate" any subset using open and closed sets, thereby defining the interior and closure.
Definition 5.4 (Interior and Closure)
Let \(A \subseteq X\) be an arbitrary subset of a topological space.
-
Interior: The interior of set \(A\) is denoted as \(A^\circ\) and is defined as the union of all open sets contained within \(A\). That is:
\[ A^\circ = \bigcup_{S \subseteq A, S \in \tau} S \]Intuitively, \(A^\circ\) is the largest open set contained entirely within \(A\).
-
Closure: The closure of set \(A\) is denoted as \(\overline{A}\) and is defined as the intersection of all closed sets containing \(A\). That is:
\[ \overline{A} = \bigcap_{A \subseteq V, V^c \in \tau} V \]Intuitively, \(\overline{A}\) is the smallest closed set containing \(A\).
Classic Examples: In the real line \(X = \mathbb{R}\) with the standard topology:
-
Let \(A = [1, 2)\), then its interior is \(A^\circ = (1, 2)\), and its closure is \(\overline{A} = [1, 2]\).
-
Let \(A = \mathbb{Q}\) (the set of rational numbers). Since the rationals contain no non-empty open intervals, its interior is \(A^\circ = \emptyset\). Since rational numbers are dense in the reals, its closure is \(\overline{A} = \mathbb{R}\).
3.1 Dual Properties of Interior and Closure
Interior and closure exhibit perfect duality via complementation operations.
Theorem 5.2 (Basic Properties and Duality)
For any subsets \(A, B\) in space \(X\), the following properties hold:
-
(1) Duality:
\[ X \setminus A^\circ = \overline{X \setminus A} \]\[ X \setminus \overline{A} = (X \setminus A)^\circ \] -
(2) Extreme Equivalence:
\[ A \text{ is an open set} \iff A = A^\circ \]\[ A \text{ is a closed set} \iff A = \overline{A} \] -
(3) Idempotency: The interior of the interior is the interior, and the closure of the closure is the closure.
\[ (A^\circ)^\circ = A^\circ \quad \text{and} \quad \overline{\overline{A}} = \overline{A} \] -
(4) Distributive Laws:
\[ (A \cap B)^\circ = A^\circ \cap B^\circ \]\[ \overline{A \cup B} = \overline{A} \cup \overline{B} \]
3.2 Boundary (Frontier) (Supplement)
With the concepts of interior and closure, we can precisely define the "edge" of a set.
Definition 5.5 (Boundary)
The boundary of set \(A\) is denoted as \(\partial A\) (or \(\text{bd}(A)\)) and is defined as the difference between the closure of the set and its interior:
Another equivalent definition of the boundary is: \(\partial A = \overline{A} \cap \overline{X \setminus A}\). This means that for any point on the boundary, every open neighborhood contains both points in \(A\) and points outside \(A\). A set is both closed and open (a "clopen" set) if and only if its boundary is empty.