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Topology 101: How Mathematicians Study Holes
- novia 6y agohumans are triple toroids. prove me wrong.
- deleted 6y ago[deleted]
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- enriquto 6y agoYou 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 https://en.wikipedia.org/wiki/Nasolacrimal_duct
- parsimo2010 6y agoI 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.
- petters 6y agoTopology 101 is not about holes. It is about open sets and continuous functions.
- enriquto 6y agoOpen 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 agoWould 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
- thearrow 6y agoIf you'd like a fun, light, video introduction to this topic, this Vsauce video is worth a watch: https://youtu.be/egEraZP9yXQ https://youtu.be/egEraZP9yXQ
- wotamRobin 6y agoWhat do topologists have for breakfast? Donuts filled with coffee.
- dsimms 6y agook, ok, the headline here is more descriptive but the one actually in the article is a good one: The Hole Truth
- peter_d_sherman 6y agoTopologist 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>
- coolgeek 6y agodupe https://news.ycombinator.com/item?id=25918577 https://news.ycombinator.com/item?id=25918577