I need to prove that if $\tau_1$ and $\tau_2$ are topologies of a set $X$, then $\tau_1 \cup \tau_2$ is not necessarily a topology on $X$.
I'm looking for counterexamples. I have one: consider $X=\{a,b,c\}$, and $\tau_1=\{\emptyset, X, \{a,c\}\}$ and $\tau_2=\{\emptyset, X, \{a,b\}\}$ then $\tau=\{\emptyset, X, \{a,c\}, \{a,b\}\}$, but note that: $\{a,b\} \cap \{a,c\} =\{a\}\notin \tau$, therefore $\tau$ is not a topology for $X$.
Is this correct? Some other more interesting counterexample?
Thanks for your help.