What is...the Zariski topology?

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VisualMath

VisualMath

Күн бұрын

Пікірлер: 9
@enstucky
@enstucky 4 ай бұрын
A quick comment on this idea of the Zariski topology not being a "real" topology. Qiachu Yuan has a fantastic pair of answers on Math StackExchange that are worth reading if someone finds this bothersome. (The question "Why Zariski topology?" by @Dubious is most relevant, and links to the other.) In brief, Yuan's thesis is that a topology in the point-set sense is better understood as a *logical* construction than a *spatial* one. The geometry-type topology is accidental in the sense that (for instance) metrics naturally ask the question "Is this there a single point that this set is close to?", and it happens that this is a question that the logical notion of a topology is equipped to understand. Clickbait title: Is Your Topologist LYING to You?? (They DO NOT Study Topologies!) (In fairness, most topologists I know wouldn't claim to- they study topological spaces :P)
@VisualMath
@VisualMath 4 ай бұрын
Great, thanks for the comment. Here is the link: math.stackexchange.com/questions/161884/why-zariski-topology And I am the lying topologist: I do not study topologies, I like the Euclidean metric 🤣
@enstucky
@enstucky 4 ай бұрын
(Sorry for the roundabout citation; just wasn't sure how you felt about links in the comments :P)
@VisualMath
@VisualMath 4 ай бұрын
@@enstucky Haha, all good. Academic links are fine 😀
@eternaldoorman5228
@eternaldoorman5228 3 ай бұрын
4:29 Your rules for a topology threw me a bit. Is this some sort of characterization as a fixed-point of a closure operator? You use V, is that V for vector space?
@eternaldoorman5228
@eternaldoorman5228 3 ай бұрын
Wikipedia's axiomatizations of Kuratowski closure are really well-written and paint a quite pretty picture. It's nice for people who like to think of these things in terms of processes rather than things that only exist somewhere out there in Plato's mind.
@VisualMath
@VisualMath 3 ай бұрын
@@eternaldoorman5228 Thanks for the question 😀 V = variety and the P and Q are the corresponding sets of polynomials. The varieties are supposed to be the closed sets of our topology, and the characterization you see is ‘‘What a topology needs to satisfy in terms of closed sets’’: proofwiki.org/wiki/Topology_Defined_by_Closed_Sets (I ignored (3), but that is trivial.) I hope that makes some sense!
@gabrielmel3972
@gabrielmel3972 9 ай бұрын
Thank you!
@VisualMath
@VisualMath 9 ай бұрын
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