# Intersection of Topologies

The intersection of any two topologies on a non empty set is always topology on that set, while the union of two topologies may not be a topology on that set.

Example:

Let $X = \left\{ {1,2,3,4} \right\}$
${\tau _1} = \left\{ {\phi ,X,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ {1,2} \right\}} \right\}$
${\tau _2} = \left\{ {\phi ,X,\left\{ 1 \right\},\left\{ 3 \right\},\left\{ {1,3} \right\}} \right\}$
${\tau _1} \cap {\tau _2} = \left\{ {\phi ,X,\left\{ 1 \right\}} \right\}$ is a topology on X.
${\tau _1} \cup {\tau _2} = \left\{ {\phi ,X,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\}\left\{ {1,3} \right\}} \right\}$ is not a topology on X.

Given two (and in fact any number of) topologies ${\tau _1}$, ${\tau _2}$ on X there is a topology $\tau = {\tau _1} \cap {\tau _2}$ which is weaker than both ${\tau _1}$ and ${\tau _2}$, is contained in both ${\tau _1}$ and ${\tau _2}$ contains every topology on X which is weaker than both ${\tau _1}$ and ${\tau _2}$.

Similarly there is a topology ${\tau ^*}$ which contains both ${\tau _1}$ and ${\tau _2}$ is the weakest in the sense that if ${\tau ^{**}}$ is a topology which contains both ${\tau _1}$ and ${\tau _2}$ then ${\tau ^*} \subseteq {\tau ^{**}}$.

We write ${\tau ^*} = \left\langle {{\tau _1},{\tau _2}} \right\rangle$ and call ${\tau ^*}$ as the topology generated by ${\tau _1}$ and ${\tau _2}$. ${\tau ^*}$ is different from the union ${\tau _1} \cup {\tau _2}$, which may not be a topology. Here ${\tau _1} \cup {\tau _2}$ is the set theoretic union of the collection ${\tau _1}$ and ${\tau _2}$.