2. Topological Spaces

Topological Spaces

Definition of a Topology on a Set

Topology: A topology on a set \( X \) is a collection \( \tau \) of subsets of \( X \) satisfying the following properties:

1. The empty set and \( X \) itself are in \( \tau \).
2. Any union of sets in \( \tau \) is in \( \tau \).
3. Any finite intersection of sets in \( \tau \) is in \( \tau \).

The elements of \( \tau \) are called open sets.

Open and Closed Sets

Open Sets: A set \( U \) in a topological space \( (X, \tau) \) is open if it belongs to the topology \( \tau \). That is, \( U \in \tau \).

Closed Sets: A set \( F \) in a topological space \( (X, \tau) \) is closed if its complement \( X \setminus F \) is open.

Basis and Subbasis for a Topology

Basis: A basis for a topology on \( X \) is a collection \( \mathcal{B} \) of subsets of \( X \) such that every open set in the topology can be written as a union of sets in \( \mathcal{B} \).

Subbasis: A subbasis for a topology on \( X \) is a collection \( \mathcal{S} \) of subsets of \( X \) such that the collection of all finite intersections of sets in \( \mathcal{S} \) forms a basis for the topology.

Examples of Topological Spaces

Discrete Topology: The discrete topology on a set \( X \) consists of all possible subsets of \( X \). That is, every subset of \( X \) is open.

Trivial Topology: The trivial topology on a set \( X \) consists only of the empty set and \( X \) itself.

Standard Topology on \( \mathbb{R} \): The standard topology on the real line \( \mathbb{R} \) is generated by the open intervals \( (a, b) \) for all \( a, b \in \mathbb{R} \).