2 ms·

Open sets lead very soon to holes, you don't even need continued functions. First you grok that a topological space is disconnected if there are more than two

Open sets lead very soon to holes, you don't even need continued functions. First you grok that a topological space is disconnected if there are more than two subsets that are open and closed. Then, a subspace has holes if its complement is disconnected.

tgb 6y ago
Would you consider a line in the plane to have holes? Its complement is disconnected. The complement of a solid torus in R^3 is not disconnected, yet it has holes in the sense discussed here. Moreover, this concept of holes is independent of a ambient set.
- enriquto 6y ago
  > Would you consider a line in the plane to have holes? Good point! I guess it's not different to the equator on the surface of a sphere... so yes, a straight line "encloses" half of the plane "inside" it, so it has a hole. Of course, my definition only works for the type of wholes that are cavities. Defining other types of holes, as discussed in the article, requires more than elementary point-set topology.
  - zonotope 6y ago
    > Good point! I guess it's not different to the equator on the surface of a sphere... so yes, a straight line "encloses" half of the plane "inside" it, so it has a hole. My topology is rusty, but I do not think that a line in the plane is equivalent to an equator of a sphere because a plane is not equivalent to a sphere in a topological sense, and neither is a line equivalent to a circle. There is no homeomorphism (structure preserving bijective function) between either pairs of spaces. There is a standard way to augment the plane/line to make it equivalent to a sphere/circle called the "1-point compactification"[1], but since it requires adding an extra point "at infinity", the augmented space is not the same as the original. So no, a straight line doesn't have a "hole" in the topological sense. The 1-point compactification of it does though. 1: https://en.wikipedia.org/wiki/Alexandroff_extension https://en.wikipedia.org/wiki/Alexandroff_extension
contravariant 6y ago
Some interesting applications of topology work on totally disconnected spaces. Sure the definition of a hole still works (as does the more proper definition) but whether it is remotely usable is a different matter.

tgb 6y ago

Would you consider a line in the plane to have holes? Its complement is disconnected. The complement of a solid torus in R^3 is not disconnected, yet it has holes in the sense discussed here. Moreover, this concept of holes is independent of a ambient set.

enriquto 6y ago
> Would you consider a line in the plane to have holes? Good point! I guess it's not different to the equator on the surface of a sphere... so yes, a straight line "encloses" half of the plane "inside" it, so it has a hole. Of course, my definition only works for the type of wholes that are cavities. Defining other types of holes, as discussed in the article, requires more than elementary point-set topology.
- zonotope 6y ago
  > Good point! I guess it's not different to the equator on the surface of a sphere... so yes, a straight line "encloses" half of the plane "inside" it, so it has a hole. My topology is rusty, but I do not think that a line in the plane is equivalent to an equator of a sphere because a plane is not equivalent to a sphere in a topological sense, and neither is a line equivalent to a circle. There is no homeomorphism (structure preserving bijective function) between either pairs of spaces. There is a standard way to augment the plane/line to make it equivalent to a sphere/circle called the "1-point compactification"[1], but since it requires adding an extra point "at infinity", the augmented space is not the same as the original. So no, a straight line doesn't have a "hole" in the topological sense. The 1-point compactification of it does though. 1: https://en.wikipedia.org/wiki/Alexandroff_extension https://en.wikipedia.org/wiki/Alexandroff_extension

contravariant 6y ago

Some interesting applications of topology work on totally disconnected spaces. Sure the definition of a hole still works (as does the more proper definition) but whether it is remotely usable is a different matter.