Definition:Generated Topology

Definition

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

Define:

$\displaystyle \mathcal S^* := \left\{{\bigcap S : S \subseteq \mathcal S \text{ finite}}\right\}$

(Note that from Intersection of Empty Set we have that $X = \bigcap \varnothing \in \mathcal S^*$.)

Then $\displaystyle \mathcal T_\mathcal S := \left\{{\bigcup C : C \subset \mathcal S^*}\right\}$ is a topology on $X$ which is said to be generated by $\mathcal S$.

Also see

• Basis

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions