nhaliday + math.gn   16

Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
nibble  qra  announcement  math  geometry  topology  puzzles  rec-math  oly  links  math.AT  ground-up  finiteness  math.GN 
october 2017 by nhaliday
general topology - What should be the intuition when working with compactness? - Mathematics Stack Exchange

The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.


As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.


Compactness does for continuous functions what finiteness does for functions in general.

If a set A is finite then every function f:A→R has a max and a min, and every function f:A→R^n is bounded. If A is compact, the every continuous function from A to R has a max and a min and every continuous function from A to R^n is bounded.

If A is finite then every sequence of members of A has a subsequence that is eventually constant, and "eventually constant" is the only kind of convergence you can talk about without talking about a topology on the set. If A is compact, then every sequence of members of A has a convergent subsequence.
q-n-a  overflow  math  topology  math.GN  concept  finiteness  atoms  intuition  oly  mathtariat  multi  discrete  gowers  motivation  synthesis  hi-order-bits  soft-question  limits  things  nibble  definition  convergence  abstraction 
january 2017 by nhaliday
Math attic
includes a nice visualization of implications between properties of topological spaces
math  visualization  visual-understanding  metabuch  techtariat  graphs  topology  synthesis  math.GN  separation  metric-space  zooming  inference  cheatsheet 
march 2016 by nhaliday

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