[HN Gopher] Topology 101: How Mathematicians Study Holes
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Topology 101: How Mathematicians Study Holes
Author : gHeadphone
Score : 67 points
Date : 2021-01-27 13:51 UTC (9 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| peter_d_sherman wrote:
| Topologist in Court: "Do you promise to tell the truth, the hole
| truth, and nothing but the truth?" <g>
|
| Rapper Ludacris as Topologist: "I got holes... in different area
| codes, area codes..." <g>
| novia wrote:
| humans are triple toroids. prove me wrong.
| parsimo2010 wrote:
| 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.
| [deleted]
| [deleted]
| enriquto wrote:
| 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.
|
| https://en.wikipedia.org/wiki/Nasolacrimal_duct
| javajosh wrote:
| 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.
| novia wrote:
| Oo very interesting. Thank you.
| thearrow wrote:
| If you'd like a fun, light, video introduction to this topic,
| this Vsauce video is worth a watch: https://youtu.be/egEraZP9yXQ
| dsimms wrote:
| ok, ok, the headline here is more descriptive but the one
| actually in the article is a good one: The Hole Truth
| wotamRobin wrote:
| What do topologists have for breakfast?
|
| Donuts filled with coffee.
| petters wrote:
| Topology 101 is not about holes. It is about open sets and
| continuous functions.
| enriquto wrote:
| 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.
| contravariant wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| > 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
| kevinventullo wrote:
| 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.
| orange_tee wrote:
| Right. This article is about what is called low-dimensional
| topology.
| spekcular wrote:
| 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.
| dwohnitmok wrote:
| 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.
| gmadsen wrote:
| maybe as a research field, but point set topology is a basic
| foundation of most mathematics
| Igelau wrote:
| > Point-set topology is as dead as the dodo.
|
| What makes it dead?
| contravariant wrote:
| 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.
| ABeeSea wrote:
| 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.
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