Give an example of topologies $\mathcal{T}$ and $\mathcal{T}'$ on $\{1,2,3\}$ such that $\mathcal{T} \cup \mathcal{T}'$ is not a topology.
By definition: A topology on a set X is a collection $\mathcal{T}$ of subsets of X having the following properties:
i) $\emptyset$ and X are in $\mathcal{T}$
ii) The union of the elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$
iii) The intersection of the elements of any finite subcollection of $\mathcal{T}$ is in $\mathcal{T}$.
On this example i'm trying to mold it with an example from the book where X=$\{a,b,c\}$ What i'm trying to understand to tackle the question, is what is the difference between the two sets that are not a topology of X, with the other sets that are topologies.
Best Answer
Just take $\mathcal{T}=\{\emptyset,\{1,2\},\{1,2,3\}\}$ and $\mathcal{T}'=\{\emptyset,\{1,3\},\{1,2,3\}\}$.
Both are topologies, but $\mathcal{T}\cup\mathcal{T}'$ is not since, for exemple, $\{1,2\}, \{1,3\}\in \mathcal{T}\cup\mathcal{T}'$ but $\{1\}=\{1,2\}\cap\{1,3\}$ is not in $\mathcal{T}\cup\mathcal{T}'$ (so ii) is not verified).