Introduction to Topology - Basics

Branch of mathematics.

Topology, a significant branch of mathematics, is often referred to as 'rubber-sheet geometry' because it involves the study of properties that remain unchanged under continuous deformations such as stretching or bending, but not tearing or gluing.

Definition of Topology

A topology on a set 'X' is a collection 'T' of subsets of 'X' that satisfies the following three properties:

1. The empty set and 'X' itself are in 'T'.
2. The intersection of any finite number of sets in 'T' is also in 'T'.
3. The union of any collection of sets in 'T' is also in 'T'.

The pair (X, T) is called a topological space.

Understanding of Topological Spaces

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

Basic Topological Properties

Open Sets and Closed Sets

In a topological space, a set is open if it belongs to the topology. The complement of an open set is called a closed set.

Interior, Closure, and Boundary

The interior of a set consists of all points in the set that can be surrounded by an open set that is entirely contained in the set. The closure of a set is the smallest closed set that contains the set. The boundary of a set is the set's closure minus its interior.

Basis for a Topology

A basis (or base) B for a topology on set X is a collection of subsets of X (called basis elements) such that every open set is a union of some basis elements.

Subspaces in Topology

A subspace of a topological space is a subset that is equipped with a topology induced from that of the larger space.

In conclusion, understanding these basic concepts is crucial for delving deeper into the subject of topology. The notions of open and closed sets, interior, closure, boundary, basis, and subspaces are fundamental to the study of topological spaces and will be used extensively as we progress through the course.