Graph (topology)

In topology, a graph is a generalization of the concept of a graph as it is known in graph theory. However, in topological contexts, the term "graph" refers to a different mathematical object, specifically a one-dimensional CW complex.

More precisely, a graph, in this topological sense, consists of a set of vertices (0-cells) and a set of edges (1-cells). Each edge is attached to one or two vertices. The vertices correspond to points, and the edges correspond to open intervals that are attached to the vertices at their endpoints. The attachment can be described by a map from the boundary of each edge (which is homeomorphic to a two-point set) to the set of vertices. This attachment may identify the two endpoints of an edge, resulting in a loop.

A graph is typically considered a topological space with the weak topology induced by the cells (vertices and edges). This means a subset is open if and only if its intersection with each vertex and each edge is open. The topological graph is therefore a Hausdorff space.

Key characteristics of topological graphs include:

• Connectivity: A graph is connected if there is a path (a sequence of edges) connecting any two vertices.
• Embeddability: A graph can be embedded into a surface if it can be drawn on the surface without any edges crossing.
• Euler Characteristic: For a finite graph, the Euler characteristic is given by V - E, where V is the number of vertices and E is the number of edges.

Topological graphs are used in various areas of mathematics, including algebraic topology, geometric topology, and knot theory. They are also related to other topological spaces, such as manifolds, and are fundamental in understanding the structure and properties of more complex spaces. The distinction between topological graphs and graphs in graph theory is crucial to avoid confusion in different mathematical contexts.