Let $X$ be a set with no topological structure but equipped with a notion of convergence. Assuming this notion of convergence satisfies some nice consistency conditions, we can define a topology on $X$ by defining an arbitrary subset $E \subset X$ to be closed if for any convergent net contained in $X$ its limit also belongs to $E$. Unfortunately, it can happen that the notion of convergence resulting from this topology might not agree with the original notion of convergence used to construct the topology. That is, if $x_\alpha \rightarrow x$ using the original notion of convergence, it might happen that $x_\alpha \rightarrow y$ in the constructed topology where $x \neq y$.
I am interested in what conditions are sufficient to ensure the two notions of convergence agree.
My motivation for this question comes from analysis where one frequently characterizes topologies by convergence. A few examples include:
- The topology of pointwise convergence
- The topology of uniform convergence
- Weak topologies
There are many ways to define these topologies. For example, the topology of pointwise can be generated using the seminorms: $$\rho_x(f) = |f(x)|$$ and the topology of uniform convergence can be generated using the supremum norm $$\|f\| = \sup_{x \in X} |f(x)|.$$ The weak topology can be generated by requiring all linear functionals to be continuous.
Could one generate these spaces using only a notion of convergence? For example take a topological vector space $X$ and as above define a topology on it using closed sets defined by weak convergence: $$\phi(f_n) \rightarrow \phi(f)$$ for all continuous linear functionals $\phi$. Would this topology agree with the usual weak topology and preserve this notion of convergence?
Similarly, if I start with a function space $X$ whose elements are real-valued and define a topology using pointwise convergence and uniform convergence would it agree with the topology of pointwise convergence and the topology of uniform convergence? Would their notions of convergence agree?
