Basic Definitions

Definition 1.7.1   Let $X$ be a set.
A topology on $X$ is $\mathcal{S}\subset \mathcal{P}(X)$ such that

1. $\mathcal{S}$ is closed under unions
2. $\mathcal{S}$ is closed under finite intersections
3. $\emptyset, X \in \mathcal{S}$. Think of elements of $\mathcal{S}$ as ``open sets"

Basic example: open sets in a metric space.

Definition 1.7.2   Ordering of topologies
If $\mathcal{S}, \mathcal{T}$ are topologies, say $\mathcal{S}$ is smaller (or weaker or coarser) if $\mathcal{S}\subset \mathcal{T}$.

The smallest topology is $\{ \emptyset, X \}$. The largest topology is $\mathcal{P}(X)$.

Given topologies $\mathcal{T}_\alpha : \alpha \in A$, there exists a unique smallest topology such that $\mathcal{T}\supset \mathcal{T}_\alpha$ and there exists a unique largest topology $\mathcal{T}$ such that $\mathcal{T}\subset \mathcal{T}_\alpha$.

Definition 1.7.3   Call $\mathcal{B}\subset \mathcal{P}(X)$ a basis if

1. $\forall x \in X, \exists B \in \mathcal{B}$ such that $x \in B$ iff $\bigcup B = X$.
2. $\forall x \in X$, $\forall B_1, B_2 \in \mathcal{B}$ such that $x \in B_1 \cap B_2$, there is $B_s \in \mathcal{B}$ such that $x \in B_3 \subset B_1 \cap B_2$ iff finite intersections of $\mathcal{B}$ can be written as unions of $B$.

Fact: If $B$ is a basis, the unions of $B$ give a topology.

Examples:

1. open intervals in $\mathbb{R}$ (usual topology).
2. $\{(a,b] : a,b \in \mathbb{R}, a < b \}$ Sorgenfriy line.
3. Order topology ($X$ with a total order). Basis: $(a,b) := \{ x \in X : a < x < b \}$, $(a,b_0]$ where $b_0$ is maximal, and $[a_0,b)$ where $a_0$ is minimal.

Definition 1.7.4   A subbasis is $\mathcal{A}\subset \mathcal{P}(X)$ where $\bigcup A = X$. The topology generated by the subbasis is the collection of unions of finite intersections of elements of $\mathcal{A}$. Equivalently, this is the smallest topology containing $\mathcal{A}$.