# 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 $${\tau _2}$$.