Existence and Uniqueness of Generated Topology

Theorem
Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a subset of the power set of $X$.

Then there exists a unique topology $\map \tau \SS$ on $X$ such that:
 * $(1): \quad \SS \subseteq \map \tau \SS$.
 * $(2): \quad$ For any topology $\TT$ on $X$, the implication $\SS \subseteq \TT \implies \map \tau \SS \subseteq \TT$ holds.

Existence
Define:
 * $\mathbb E = \leftset {\TT \subseteq \powerset X: \SS \subseteq \TT}$ and $\TT$ is a topology on $\rightset X$

Since $\powerset X$ is a topology on $X$, it follows that $\mathbb E$ is non-empty.

Hence, we can define:
 * $\ds \map \tau \SS = \bigcap \mathbb E$

It follows that $\map \tau \SS$ is a topology on $X$.

By Intersection is Largest Subset: General Result, it follows that $\SS \subseteq \map \tau \SS$.

By Intersection is Subset: General Result, it follows that if $\SS \subseteq \TT$ and $\TT$ is a topology on $X$, then $\map \tau \SS \subseteq \TT$.

Uniqueness
Suppose that $\TT_1$ and $\TT_2$ are both topologies on $X$ satisfying conditions $(1)$ and $(2)$.

By condition $(1)$, we have $\SS \subseteq \TT_2$; hence, we can apply condition $(2)$ to conclude that:
 * $\TT_1 \subseteq \TT_2$

Similarly:
 * $\TT_2 \subseteq \TT_1$

By definition of set equality:
 * $\TT_1 = \TT_2$

Also see

 * Definition:Topology Generated by Synthetic Sub-Basis