Topology

Basic Concepts

Topology Space

A topology space is a set \(X\) with a collection of open sets \(\mathcal{T}\) that satisfy the following properties: 1. \(\emptyset \in \mathcal{T}\) and \(X \in \mathcal{T}\). 2. if \(\{U_i\}_{i \in I} \subseteq \mathcal{T}\), then \(\bigcup_{i \in I} U_i \in \mathcal{T}\). 3. if \(U_1, U_2 \in \mathcal{T}\), then \(U_1 \cap U_2 \in \mathcal{T}\).

A topological space is denoted as \((X, \mathcal{T})\).

Basis and subbasis for a Topology

A basis \(\mathcal{B}\) for a topology on \(X\) is a collection of sets such that: 1. For each \(x \in X\), there exists a basis element \(B \in \mathcal{B}\) such that \(x \in B\). 2. If \(B_1, B_2 \in \mathcal{B}\) and \(x \in B_1 \cap B_2\), then there exists a basis element \(B_3 \in \mathcal{B}\) such that \(x \in B_3 \subseteq B_1 \cap B_2\).

A topology generated by a basis \(\mathcal{B}\) is the collection of all unions of elements of \(\mathcal{B}\). Formally, \(\mathcal{T} = \{ U \subseteq X : U = \bigcup_{i \in I} B_i, B_i \in \mathcal{B}, I \text{ is an index set} \}\).

A subbasis for a topology on \(X\) is a collection of sets \(\mathcal{S}\) such that the collection of finite intersections of elements of \(\mathcal{S}\) forms a basis for a topology on \(X\). Formally, a subbasis \(\mathcal{S}\) satisfies: 1. For each \(x \in X\), there exists a subbasis element \(S \in \mathcal{S}\) such that \(x \in S\). 2. The collection of finite intersections of elements of \(\mathcal{S}\) generates a basis for the topology on \(X\).

A topology generated by a subbasis \(\mathcal{S}\) is denoted as \(\mathcal{T} = \{ U \subseteq X : U = \bigcup_{i \in I} \bigcap_{j=1}^{n_i} S_{ij}, S_{ij} \in \mathcal{S}, n_i \text{ is finite}, I \text{ is an index set} \}\).

Countability, Compactness and Separation Axioms

A topological space is first countable if every point has a countable local base, meaning for each point \(x \in X\), there exists a countable collection of open sets \(\{U_n\}_{n=1}^{\infty}\) such that for any open set \(U\) containing \(x\), there exists some \(n\) such that \(U_n \subseteq U\).

A topological space is second countable if it has a countable base, meaning there exists a countable collection of open sets \(\mathcal{B}\) such that every open set in the topology can be expressed as a union of sets from \(\mathcal{B}\).

A topological space is compact if every open cover has a finite subcover, meaning for any collection of open sets \(\{U_i\}_{i \in I}\) such that \(X = \bigcup_{i \in I} U_i\), there exists a finite subset \(J \subseteq I\) such that \(X = \bigcup_{j \in J} U_j\).

A topological space is \(T_0\) (Kolmogorov) if for any two distinct points \(x, y \in X\), there exists an open set containing one of the points but not the other. Intuitively, this means that a pair of point can be one-sidely distinguished by open sets for one of them.

A topological space is \(T_1\) (Frechet) if for any two distinct points \(x, y \in X\), there exist open sets \(U_x\) and \(U_y\) such that \(x \in U_x\) and \(y \notin U_x\), and \(y \in U_y\) and \(x \notin U_y\). Intuitively, this means that points can be one-sidely separated by open sets for both of them.

A topological space is \(T_2\) (Hausdorff) if for any two distinct points \(x, y \in X\), there exist disjoint open sets \(U_x\) and \(U_y\) such that \(x \in U_x\) and \(y \in U_y\). This means that points can be separated by disjoint open sets.

A topological space is regular if it is \(T_1\) and for every point \(x\) and closed set \(F\) not containing \(x\), there exist disjoint open sets \(U\) and \(V\) such that \(x \in U\) and \(F \subseteq V\). This means that points can be separated from closed sets by disjoint open sets.

A topological space is normal if for any two disjoint closed sets \(A\) and \(B\), there exist disjoint open sets \(U\) and \(V\) such that \(A \subseteq U\) and \(B \subseteq V\). This means that closed sets can be separated by disjoint open sets.

A topological space is \(T_3\) (regular Hausdorff) if it is both regular and Hausdorff. This means that points can be separated from closed sets by disjoint open sets, and distinct points can be separated by disjoint open sets. \(T_3\) spaces are also normal.

Important properties of these separation axioms include: - \(T_0 \subseteq T_1 \subseteq T_2 \subseteq T_3\). - A single point set is closed in a \(T_1\) space. - A \(T_2\) space is normal if it is also regular since a point is closed in a \(T_2\) space.

Functions and Homeomorphisms

A function \(f: X \to Y\) between two topological spaces \((X, \mathcal{T}_X)\) and \((Y, \mathcal{T}_Y)\) is open if for every open set \(U \in \mathcal{T}_X\), the image \(f(U)\) is open in \(\mathcal{T}_Y\). In other words, \[ f(U) \in \mathcal{T}_Y \text{ for all } U \in \mathcal{T}_X.\]

A function \(f: X \to Y\) is closed if for every closed set \(C \in \mathcal{T}_X\), the image \(f(C)\) is closed in \(\mathcal{T}_Y\). In other words, \[ f(C) \in \mathcal{T}_Y \text{ for all } C \in \mathcal{T}_X.\]

A function \(f: (X, \mathcal{T}_X) \to (Y, \mathcal{T}_Y)\) between topological spaces is continuous if for every open set \(V \in \mathcal{T}_Y\), the preimage \(f^{-1}(V)\) is open in \(\mathcal{T}_X\). In other words, \[ f^{-1}(V) \in \mathcal{T}_X \text{ for all } V \in \mathcal{T}_Y.\]

A function \(f: (X, \mathcal{T}_X) \to (Y, \mathcal{T}_Y)\) is a homeomorphism if it is a continuous bijection and its inverse \(f^{-1}: (Y, \mathcal{T}_Y) \to (X, \mathcal{T}_X)\) is also continuous. In other words, both \(f\) and \(f^{-1}\) are continuous functions.

A homeomorphism establishes a topological equivalence between the two spaces, meaning they have the same topological properties.

Connectedness and Path Connectedness

A topological space is connected if it cannot be expressed as the union of two disjoint non-empty open sets. In other words, there are no two open sets \(U\) and \(V\) such that \(X = U \cup V\), \(U \cap V = \emptyset\), and both \(U\) and \(V\) are non-empty.

A topological space is path connected if for any two points \(x, y \in X\), there exists a continuous function (path) \(f: [0, 1] \to X\) such that \(f(0) = x\) and \(f(1) = y\). This means that there is a continuous path connecting any two points in the space.

A topological space is locally path connected if every point has a neighborhood base consisting of path connected sets. This means that for each point \(x \in X\), there exists a collection of open sets \(\{U_n\}_{n=1}^{\infty}\) such that for each \(n\), \(U_n\) is path connected and contains \(x\), and for any open set \(U\) containing \(x\), there exists some \(n\) such that \(U_n \subseteq U\).

A topological space is locally connected if every point has a neighborhood base consisting of connected sets. This means that for each point \(x \in X\), there exists a collection of open sets \(\{U_n\}_{n=1}^{\infty}\) such that for each \(n\), \(U_n\) is connected and contains \(x\), and for any open set \(U\) containing \(x\), there exists some \(n\) such that \(U_n \subseteq U\).

Common Topologies

Fineness and coarseness of topologies are important concepts in topology. A topology \(\mathcal{T}_1\) is finer than another topology \(\mathcal{T}_2\) on the same set \(X\) if every open set in \(\mathcal{T}_2\) is also an open set in \(\mathcal{T}_1\). In other words, \(\mathcal{T}_1 \supseteq \mathcal{T}_2\). Conversely, \(\mathcal{T}_2\) is coarser than \(\mathcal{T}_1\) if \(\mathcal{T}_1\) is finer than \(\mathcal{T}_2\).

The discrete topology on a set \(X\) is the finest topology, where every subset of \(X\) is open. Formally, \(\mathcal{T}_{\text{discrete}} = \mathcal{P}(X)\), the power set of \(X\).

The indiscrete topology (or trivial topology) on a set \(X\) is the coarsest topology, where only the empty set and the entire set \(X\) are open. Formally, \(\mathcal{T}_{\text{indiscrete}} = \{\emptyset, X\}\).

The quotient topology on a set \(X\) with an equivalence relation \(\sim\) is defined as follows: 1. The quotient set is \(X / \sim = \{ [x] : x \in X \}\), where \([x]\) is the equivalence class of \(x\). 2. A subset \(U \subseteq X / \sim\) is open in the quotient topology if and only if its preimage under the natural projection map \(\pi: X \to X / \sim\) is open in \(X\). That is, \(U\) is open if \(\pi^{-1}(U)\) is open in \(X\).

In other words, the quotient topology is the coarsest topology on \(X / \sim\) such that the natural projection map \(\pi\) is continuous.

The product topology on a product of topological spaces \(\{X_i\}_{i \in I}\) is defined as follows: 1. The product space is \(X = \prod_{i \in I} X_i\). 2. A subset \(U \subseteq X\) is open in the product topology if it can be expressed as a union of sets of the form \(\prod_{i \in I} U_i\), where \(U_i\) is open in \(X_i\) for each \(i \in I\), and \(U_i = X_i\) for all but finitely many \(i\). In other words, the product topology is generated by the basis consisting of all products of open sets, where only finitely many factors are not the entire space.

The box topology on a product of topological spaces \(\{X_i\}_{i \in I}\) is defined similarly to the product topology, but without the restriction that only finitely many factors can be different from the entire space: 1. The box space is \(X = \prod_{i \in I} X_i\). 2. A subset \(U \subseteq X\) is open in the box topology if it can be expressed as a union of sets of the form \(\prod_{i \in I} U_i\), where \(U_i\) is open in \(X_i\) for each \(i \in I\). In other words, the box topology is generated by the basis consisting of all products of open sets, where each factor can be any open set in the corresponding space.

The subspace topology on a subset \(Y \subseteq X\) of a topological space \((X, \mathcal{T})\) is defined as follows: 1. The subspace is \(Y\). 2. A subset \(U \subseteq Y\) is open in the subspace topology if it can be expressed as \(U = Y \cap V\) for some open set \(V \in X\). In other words, \(U\) is open in the subspace topology if it is the intersection of \(Y\) with an open set in \(X\).