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.
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.
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.
> 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.
> 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.
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.
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.
I dont think so. Topology is ubiquitous in mathematics. From Zariski topologies and
Stone spaces, to concrete objects like manifolds and topological vector spaces.
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.
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.
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.
You have two nasolacrimal ducts that connect your nose to your eyes. Moreover, the upper end of the tear duct has two openings. Thus you have at least seven holes.
Sure but the ear/nose/throat are basically just a filter of the mouth hole. But yeah humans are basically a GI tract plus appendages, which is indeed topologically a toroid.
I think it depends on what "counts" as a hole. Presumably you're counting your digestive tract as a hole from the mouth to the anus. But by virtue of me being able to hold my farts in, my anus is airtight. I don't think that the digestive tract should count as a hole when my anus is in the normal airtight state. I think that normally I'm a double toroid and I temporarily become a triple toroid when someone pulls my finger.
