This section introduces the concept of topological spaces, which have roots in geometry and analysis. Topology studies properties preserved under homeomorphisms and abstracts classical concepts from real or complex function analysis, such as open sets, continuity, connectedness, compactness, and metric spaces. A topological space is defined as a set X with a collection T of subsets (called open sets) satisfying three conditions: the empty set and X are in T, the union of any collection of open sets is open, and the intersection of any two open sets is open. Examples include the usual topology on real numbers and the plane, as well as discrete and indiscrete topologies. A base for a topology is a collection of open sets such that every open set can be expressed as a union of base elements. The relative topology on a subset A of a topological space X is defined by intersecting open sets of X with A. An open neighborhood of a point x is an open set containing x. A limit point of a subset A is a point where every open neighborhood of x intersects A nontrivially. The derived set of A is the set of all its limit points. The section also provides examples of topological spaces and discusses the properties of open sets and neighborhoods.This section introduces the concept of topological spaces, which have roots in geometry and analysis. Topology studies properties preserved under homeomorphisms and abstracts classical concepts from real or complex function analysis, such as open sets, continuity, connectedness, compactness, and metric spaces. A topological space is defined as a set X with a collection T of subsets (called open sets) satisfying three conditions: the empty set and X are in T, the union of any collection of open sets is open, and the intersection of any two open sets is open. Examples include the usual topology on real numbers and the plane, as well as discrete and indiscrete topologies. A base for a topology is a collection of open sets such that every open set can be expressed as a union of base elements. The relative topology on a subset A of a topological space X is defined by intersecting open sets of X with A. An open neighborhood of a point x is an open set containing x. A limit point of a subset A is a point where every open neighborhood of x intersects A nontrivially. The derived set of A is the set of all its limit points. The section also provides examples of topological spaces and discusses the properties of open sets and neighborhoods.