6 ms·

Topology 101 is not about holes. It is about open sets and continuous functions.

enriquto 6y ago
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.
orange_tee 6y ago
Right. This article is about what is called low-dimensional topology.
ABeeSea 6y ago
So after the first 5 minutes, what do you spend the rest of topology doing? Abstract point set topology came way after the study of “holes” chronologically. Riemann used the genus an an invariant of compact surfaces when he was studying complex integrals.
kevinventullo 6y ago
Yes! There are applications of “Topology 101” that are not Euclidean spaces, e.g. “Totally disconnected” spaces show up early on in number theory, and the Zariski topology associated to an algebraic variety/scheme is foundational in algebraic geometry. That said, any mathematician calling themselves a “topologist” is likely studying things like homology/homotopy or generalizations which mostly apply to Euclideanish spaces.
spekcular 6y ago
I guess it's worth pointing out that when mathematicians today use the word "topology" in conversation, they almost universally mean "algebraic topology." Point-set topology is as dead as the dodo.
- Igelau 6y ago
  > Point-set topology is as dead as the dodo. What makes it dead?
  - spekcular 6y ago
    No one publishes on it anymore. My impression is that all the interesting problems are solved. I also can't think of any professor at, say, an R1 university that works primarily on point-set topology.
- gmadsen 6y ago
  maybe as a research field, but point set topology is a basic foundation of most mathematics
- dwohnitmok 6y ago
  Pure point-set topology is as "dead" as group theory. That is to say it's an essential part of the foundations of theoretical mathematics (you can't understand algebraic topology without understanding point-set topology), but there is not much new innovation purely within point-set topology itself. So I find "dead as a dodo" too strong of a metaphor.
- contravariant 6y ago
  Where on earth did you get that idea? The most common meaning of the word "topology" is quite simply "a topology". As in a set of opens for a particular space of interest. These show up all over the place regardless of whether you can do some algebra on them.
  - spekcular 6y ago
    I meant as a field of research. Obviously the concept of an open set, etc., remain foundational.

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.

orange_tee 6y ago

Right. This article is about what is called low-dimensional topology.

ABeeSea 6y ago

So after the first 5 minutes, what do you spend the rest of topology doing? Abstract point set topology came way after the study of “holes” chronologically. Riemann used the genus an an invariant of compact surfaces when he was studying complex integrals.

kevinventullo 6y ago

Yes! There are applications of “Topology 101” that are not Euclidean spaces, e.g. “Totally disconnected” spaces show up early on in number theory, and the Zariski topology associated to an algebraic variety/scheme is foundational in algebraic geometry. That said, any mathematician calling themselves a “topologist” is likely studying things like homology/homotopy or generalizations which mostly apply to Euclideanish spaces.

spekcular 6y ago

I guess it's worth pointing out that when mathematicians today use the word "topology" in conversation, they almost universally mean "algebraic topology." Point-set topology is as dead as the dodo.

Igelau 6y ago
> Point-set topology is as dead as the dodo. What makes it dead?
- spekcular 6y ago
  No one publishes on it anymore. My impression is that all the interesting problems are solved. I also can't think of any professor at, say, an R1 university that works primarily on point-set topology.
gmadsen 6y ago
maybe as a research field, but point set topology is a basic foundation of most mathematics
dwohnitmok 6y ago
Pure point-set topology is as "dead" as group theory. That is to say it's an essential part of the foundations of theoretical mathematics (you can't understand algebraic topology without understanding point-set topology), but there is not much new innovation purely within point-set topology itself. So I find "dead as a dodo" too strong of a metaphor.
contravariant 6y ago
Where on earth did you get that idea? The most common meaning of the word "topology" is quite simply "a topology". As in a set of opens for a particular space of interest. These show up all over the place regardless of whether you can do some algebra on them.
- spekcular 6y ago
  I meant as a field of research. Obviously the concept of an open set, etc., remain foundational.