Regarding your point 3 here is a list of some sources you might find interesting:

• Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about
• Schechter, "Hadbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way
• Nel, "Continuity Theory"
• Dolecki, "An invitation into Convergence Theory"
• Dolecki, "A Royal Road to Topology: Convergence of Filters"

You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".

As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of, filter bases to be more precise): Zorich "Mathematical Analysis".

Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.

There's even more general notion, see Joseph Muscat "An Axiomatization of Filter Clustering".

Regarding your point 3 here is a list of some sources you might find interesting:

• Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about
• Schechter, "Handbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way
• Nel, "Continuity Theory"
• Dolecki, "An invitation into Convergence Theory"
• Dolecki, "A Royal Road to Topology: Convergence of Filters"

You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".

As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of, filter bases to be more precise): Zorich "Mathematical Analysis".

Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.

There's even more general notion, see Joseph Muscat "An Axiomatization of Filter Clustering".

Regarding your point 3 here is a list of some sources you might find interesting:

• Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about
• Schechter, "Hadbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way
• Nel, "Continuity Theory"
• Dolecki, "An invitation into Convergence Theory"

You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".

As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of): Zorich "Mathematical Analysis".

Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.

Regarding your point 3 here is a list of some sources you might find interesting:

• Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about
• Schechter, "Hadbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way
• Nel, "Continuity Theory"
• Dolecki, "An invitation into Convergence Theory"
• Dolecki, "A Royal Road to Topology: Convergence of Filters"

You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".

As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of, filter bases to be more precise): Zorich "Mathematical Analysis".

Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.

There's even more general notion, see Joseph Muscat "An Axiomatization of Filter Clustering".