Intersection of Topologies

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.


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}.