Existence and Uniqueness of Generated Topology

Theorem
Let $X$ be a set.

Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a subset of the power set of $X$.

Then there exists a unique topology $\tau \left({\mathcal S}\right)$ on $X$ such that:
 * $\left({1}\right): \quad$ $\mathcal S \subseteq \tau \left({\mathcal S}\right)$.
 * $\left({2}\right): \quad$ For any topology $\mathcal T$ on $X$, the implication $\mathcal S \subseteq \mathcal T \implies \tau \left({\mathcal S}\right) \subseteq \mathcal T$ holds.

Existence
Define:
 * $\mathbb E = \left\{{\mathcal T \subseteq \mathcal P \left({X}\right): \mathcal S \subseteq \mathcal T}\right.$ and $\mathcal T$ is a topology on $\left.{X}\right\}$

Since $\mathcal P \left({X}\right)$ is a topology on $X$, it follows that $\mathbb E$ is non-empty.

Hence, we can define:
 * $\displaystyle \tau \left({\mathcal S}\right) = \bigcap \mathbb E$

It follows that $\tau \left({\mathcal S}\right)$ is a topology on $X$.

By Intersection Largest: General Result, it follows that $\mathcal S \subseteq \tau \left({\mathcal S}\right)$.

By Intersection Subset: General Result, it follows that if $\mathcal S \subseteq \mathcal T$ and $\mathcal T$ is a topology on $X$, then $\tau \left({\mathcal S}\right) \subseteq \mathcal T$.

Uniqueness
Suppose that $\mathcal T_1$ and $\mathcal T_2$ are both topologies on $X$ satisfying conditions $\left({1}\right)$ and $\left({2}\right)$.

By condition $\left({1}\right)$, we have $\mathcal S \subseteq \mathcal T_2$; hence, we can apply condition $\left({2}\right)$ to conclude that $\mathcal T_1 \subseteq \mathcal T_2$.

Similarly, $\mathcal T_2 \subseteq \mathcal T_1$.

Therefore, by Equality of Sets, it follows that $\mathcal T_1 = \mathcal T_2$.

Also see
For an alternative proof of existence, see Equivalence of Definitions of Generated Topology.


 * Generated Topology