Topology Generated by a Set

Last updated Nov 1, 2022

Let $X$ be a Set and let $B \subset \mathcal{P}(X)$. Then we define $\tau(B)$, the Topology Generated by a Set as $$\tau(B) := \bigcap\limits {\tau \subset \mathcal{P}(X) : B \subset \tau, \tau \text{ is a topology on }X}.$$ In other words, $\tau(B)$ is the “smallest” Topological Space containing $B$.

1. $\tau(B)$ is a Topological Space since Intersection of Structures is still a Structure.

• Topological Space