Declension wrote:
Tuttle wrote:
Point-set topology, algebraic topology, or something different?
Point-set. I am currently writing a master's thesis on inverse limits in a certain category, where the category has object class "topological spaces" and morphism class "upper hemicontinuous set-valued maps".
http://en.wikipedia.org/wiki/Inverse_limithttp://en.wikipedia.org/wiki/HemicontinuityThe categorical definition of "inverse limit" looks a bit scary, but it is actually quite intuitive once you translate everything into topologese. It feels like you're following points on their infinite journey backwards through a maze of mappings.
I don't see the intuitive nature, but I can see how its useful and can somewhat see how it would be. That sounds interesting

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Quote:
Tuttle wrote:
Specifically where I want to go is I want to get a PhD in mathematical logic
I like logic too. I like the idea that there is actually a formal account of what mathematicians are supposed to be doing, although it sometimes doesn't feel like that's what mathematicians are really doing. Sometimes I like to think about how topology would be affected if the foundations of mathematics were different. There's a completely central result in point-set topology which is actually equivalent (in ZF) to the Axiom of Choice!

The relationship between logic and topology is quite interesting to me. I really enjoy the fact that first order logic is compact. As an undergrad I learned the definition of compactness in three different class-like things in three different fields (analysis, logic, and topology - though only one was in an undergrad class), and they all were the same definition just in the language of that field. I really enjoy those types of relationships between fields of mathematics.
Last edited by Tuttle on 20 Mar 2012, 11:26 am, edited 1 time in total.