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 $a,b\in \mathbb{R}$.