I will give a little background. I have just started a course on differential geometry and prior to this, I had only about vector spaces. I had not read a single thing about topology or topological spaces or anything such. So prior to taking this course, I only knew meaning of open sets in the context of vector spaces.
In the beginning of this course, all we were taught about topology is the axioms that a set of subsets of a parent set should satisfy. And that exact set of a topology is an open set. Now I am having trouble in reconciling the idea of open sets in both contexts (one being vector spaces and other being topologies).
My problem arises because in the definition of charts of a non empty set, it is required that the image of chart map is an open set in $R^n$. So what does open set mean here? In lecture notes they do mention something like whenever $R^n$ is considered as a vector space, it is equipped with its standard topology. What is a standard topology (I googled it but all the sources assume different backgrounds and it would require me to again read something else hence I ask here)? And does it mean that any open set in the vector space sense is also open in the context of its standard topology and vice versa?