Generating topological space diagram in Mathematica

I have mathematica code to check whether a collection of sets satisfies the definition of a topology, I would now like to programmatically generate diagrams like these: topological spaces

How can this be done?

Solution

I'm not familiar with your problem but to create diagrams from primitives, that look kind of like the ones you have pasted, you can do this:

start with the "base" case --

base = {Circle[{-0.4, 0.4}, 0.1], Disk[{0, .125}, 0.05], 
   Text[Style["1", 24], {0, -0.1}],
   Disk[{0.5, .125}, 0.05], Text[Style["2", 24], {0.5, -0.1}], 
   Disk[{1., .125}, 0.05], Text[Style["3", 24], {1., -0.1}], 
   Circle[{.5, 0}, {.9, .5}]};

Graphics[{base}, ImageSize -> 220]

enter image description here

From here just add elipses to the base case:

Graphics[{base, Circle[{0, 0}, {.15, .3}]}, ImageSize -> 220]

enter image description here

Graphics[{base, Circle[{0, 0}, {.15, .3}], 
  Circle[{0.5, 0}, {.15, .3}], Circle[{0.25, 0}, {.58, .38}]}, 
 ImageSize -> 220]

enter image description here

Graphics[{base, Circle[{0.5, 0}, {.15, .3}], 
  Circle[{0.25, 0}, {.58, .38}], Circle[{0.75, 0}, {.58, .38}]}, 
 ImageSize -> 220]

enter image description here

Graphics[{base, Circle[{0.5, 0}, {.15, .3}], 
  Circle[{1, 0}, {.15, .3}], Red, AbsoluteThickness[6], 
  Line[{{-0.4, -0.5}, {1.4, 0.55}}], 
  Line[{{-0.4, 0.55}, {1.4, -0.5}}]}, ImageSize -> 220]

enter image description here

Graphics[{base, Circle[{0.25, 0}, {.58, .38}], 
  Circle[{0.75, 0}, {.58, .38}], Red, AbsoluteThickness[6], 
  Line[{{-0.4, -0.5}, {1.4, 0.55}}], 
  Line[{{-0.4, 0.55}, {1.4, -0.5}}]}, ImageSize -> 220]

enter image description here

Note that I set Frame->True while tweaking these so I could see the coordinates.