How far is this view correct on strengthening and weakening of the topologies of the domain and the codomain? Let $f:X \to Y$ be a map between infinite topological spaces. I want to know how correct is this view on how refining (strengthening) and coarsening (weakening) the topologies on both the domain and the codomain affects a mapping being continuous and open mapping.
Question: Is there in general such diagram, (or a similar one,) that holds true whenever one starts from the trivial topologies and refines them (taking anyway) towards the discrete topologies ?


Starting from the trivial (chaotic) topologies, there are too many paths toward the discrete topologies, not only one, but is there a diagram of that sort above that holds no matter what, i.e. will change "homeomorphically" under changing the path of refinment.
Feel free to suggest corrections and diagrams.
 A: Note: this answer is not entirely applicable anymore, as the question has changed.
Your diagram implies that if you have any continuous non-open map between $X$ and $Y$, it is always sufficient to weaken the topology of $X$ to make the map continuous and open. This is simply not true in general.

Example: Take $X=[0,+\infty)$, $Y=\mathbb R$, and $f(x)=x$ being the inclusion map. Consider on $Y$ the Euclidean topology and on $X$ the topology induced by the map $f$, i.e., the weakest topology that makes $f$ continuous (this induced topology on $X$ coincides in turn with the Euclidean topology, as the topology induced by the inclusion map is just the subspace topology). The map is then continuous but not open, as the image of $X$ is $[0,+\infty)\subset Y$. If you then weaken the topology of $X$ to try to make $f$ open, you necessarily end up making it discontinuous.
If you instead strengthen the topology of $X$, you necessarily have a non-open map, as “you have even more sets to be tested for openness”. You can play around with this example and similar ones to see how and where the diagram fails. I think an interesting way is to fix one of the two topologies and let the other one vary starting from the topology induced by the map $f$, which should correspond to the treshold you drew between the “continuous” and “non-continuous” regions.
