This section introduces the fundamental concepts of topological spaces, which are rooted in geometry and analysis. Topology studies properties preserved by homeomorphisms and abstracts classical concepts such as open sets, continuity, connectedness, compactness, and metric spaces. The key definition is that of a topology on a set \(X\), which is a collection \(T\) of subsets of \(X\) satisfying three conditions: the empty set and \(X\) are in \(T\), any union of members of \(T\) is in \(T\), and any intersection of two members of \(T\) is in \(T\). The members of \(T\) are called open sets, and the pair \((X, T)\) is a topological space. Examples include the usual topology on \(\mathbb{R}\) and \(\mathbb{R}^2\) formed by unions of open intervals and open spheres, respectively. Other examples illustrate that not all collections of subsets satisfy the conditions to be a topology, such as \(\{X, \emptyset, \{x\}, \{z, u\}, \{x, z, u\}, \{y, z, u\}, \{y, z, u, v\}\}\) for the set \(\{x, y, z, u, v\}\). The discrete topology and indiscrete topology are also discussed, where every subset or only \(\{X, \emptyset\}\) is open, respectively. Additionally, a base for a topology is defined, and it is shown that open intervals form a base for the usual topology on \(\mathbb{R}\). The section also introduces the concept of the relative topology on a subset \(A\) of \(X\), denoted by \(T_A\), and defines limit points and derived sets in a topological space.This section introduces the fundamental concepts of topological spaces, which are rooted in geometry and analysis. Topology studies properties preserved by homeomorphisms and abstracts classical concepts such as open sets, continuity, connectedness, compactness, and metric spaces. The key definition is that of a topology on a set \(X\), which is a collection \(T\) of subsets of \(X\) satisfying three conditions: the empty set and \(X\) are in \(T\), any union of members of \(T\) is in \(T\), and any intersection of two members of \(T\) is in \(T\). The members of \(T\) are called open sets, and the pair \((X, T)\) is a topological space. Examples include the usual topology on \(\mathbb{R}\) and \(\mathbb{R}^2\) formed by unions of open intervals and open spheres, respectively. Other examples illustrate that not all collections of subsets satisfy the conditions to be a topology, such as \(\{X, \emptyset, \{x\}, \{z, u\}, \{x, z, u\}, \{y, z, u\}, \{y, z, u, v\}\}\) for the set \(\{x, y, z, u, v\}\). The discrete topology and indiscrete topology are also discussed, where every subset or only \(\{X, \emptyset\}\) is open, respectively. Additionally, a base for a topology is defined, and it is shown that open intervals form a base for the usual topology on \(\mathbb{R}\). The section also introduces the concept of the relative topology on a subset \(A\) of \(X\), denoted by \(T_A\), and defines limit points and derived sets in a topological space.