nhaliday + math.at   12

gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.


Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
nibble  q-n-a  overflow  math  geometry  topology  tidbits  intricacy  intersection  proofs  gotchas  oly  mathtariat  fixed-point  math.AT  manifolds  intersection-connectedness 
october 2017 by nhaliday
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
Covering space - Wikipedia
A covering space of X is a topological space C together with a continuous surjective map p: C -> X such that for every x ∈ X, there exists an open neighborhood U of x, such that p^−1(U) (the inverse image of U under p) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.
concept  math  topology  arrows  lifts-projections  wiki  reference  fiber  math.AT  nibble  preimage 
january 2017 by nhaliday

bundles : academeframemathsp

related tags

advanced  algebra  algorithms  announcement  aphorism  applications  arrows  automation  biases  big-list  big-picture  bio  books  concentration-of-measure  concept  concrete  cs  current-events  data-science  degrees-of-freedom  differential  electromag  elegance  estimate  ethical-algorithms  examples  existence  extrema  fiber  finiteness  fixed-point  fourier  geography  geometry  gotchas  government  graphs  ground-up  heterodox  hmm  init  inner-product  insight  interdisciplinary  intersection  intersection-connectedness  interview  intricacy  latex  law  lifts-projections  links  list  local-global  manifolds  maps  math  math.AT  math.CA  math.CO  math.CT  math.FA  math.GN  math.MG  mathtariat  meta:math  motivation  multi  news  nibble  oly  orders  org:edu  org:inst  org:junk  org:mag  org:sci  overflow  p:***  p:whenever  pennsylvania  physics  polisci  politics  polynomials  preimage  probabilistic-method  profile  proofs  puzzles  q-n-a  qra  quantifiers-sums  quixotic  rec-math  reference  s:**  social-choice  soft-question  survey  symmetry  synchrony  synthesis  tcs  teaching  tidbits  tools  topics  topology  tricki  tricks  uniqueness  unit  usa  vague  visual-understanding  volo-avolo  wiki  wisdom  wormholes  yak-shaving  yoga  👳 

Copy this bookmark: