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March 1

Philosophical question about Zariski and metric topologies

I accidentally asked this on science area first, sorry. How do the Zariski and metric topologies on the complex numbers interact or complement each other when mathematicians are studying algebraic geometry or several complex variables or in other areas of mathematics? Thanks. Rich (talk) 20:43, 1 March 2025 (UTC)[reply]

I guess the basic answer is that the metric topology (or analytic topology) is much stronger than the Zariski topology, and therefore more "intuitive". However, in many cases there are deep connections between the two topologies (e.g., Serre's GAGA and Chow's theorem). Tito Omburo (talk) 21:22, 1 March 2025 (UTC)[reply]

Ok thank you. it's heavy reading but i'll tackle it. One question left is : is metric topology ALWAYS strictly finer than Zariski that every open set in Zariski is also always open in metric topology? Because in several complex variables zero sets(the closed sets) don't have to be isolated if I remember rightly.Rich (talk) 06:59, 2 March 2025 (UTC)[reply]

It's always strictly finer because the Zariski topology is defined by polynomial functions, which are continuous in the metric topology. Tito Omburo (talk) 10:50, 2 March 2025 (UTC)[reply]