Give an example of a space $X$, a subset $A$ and topology on $A$ such that $i: A \rightarrow X$ is NOT continuous I was thinking of taking $X=\\{0,1,2\\}$ and $A=\\{1,2\\}$ And then the primage of ${0}$ is empty? I do not know which topology to take Thanks

Take $X=\mathbb R$ with usual topology and $A=\mathbb Z$ with indiscrete topology. (The one in which only empty set and whole set are open). This is not continuous as the inverse image $i^{-1}(n-\epsilon,n+\epsilon)=\\{n\\}$ for any $n\in \mathbb Z$ and for some $0<\epsilon <1$ is not open in $\mathbb Z$ and hence $i$ is not continuous.