# 8

# Basic Reminders in Topology and Functional Analysis

Topology and Functional Analysis are of key importance in Mathematical Imaging because they provide a lot of notions and tools that will be useful for setting up and studying the mathematical frameworks for both gray-tone and binary images.

# 8.1. Mathematical disciplines

The basic mathematical discipline is **Topology** [KEL 75, JÄN 84], which was historically developed from the concepts arising from Geometry and Set Theory. It deals with abstract ‘spatial’ properties, primarily for continuity based on the central concept of neighborhood. Topological spaces are mathematical structures that make it possible to formally define concepts such as continuity, convergence, compactness and connectedness. They appear in virtually every branch of modern Mathematics and are a central unifying notion.

The other main mathematical discipline of reference is **Functional Analysis** [KOL 99; Original ed., 1954 and 1957] [KAN 82] [KRE 89; 1st ed., 1978] that deals with the study of functions in concrete or abstract forms.

# 8.2. Topological spaces

## 8.2.1. *Neighborhood systems*

A **topological space** is a set in which is defined, for any of its elements , a so-called fundamental system of neighborhoods, or for short a **neighborhood system**, which consists of assigning ...