TOPOLOGY
Topology = Idealization of space
euclidean
noneuclidean
analytic
affine
projective geometry
Klein-Erlanger Program -> Transformation key! not object!
1. Determine topological equivalence of spaces (Homeomorphics)
2. Determine if space x has fixed point
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Topology
Classify Topology spaces - equivalence classes/cannonical instances
1. Spheres w/ g (genus) handles (All closed surface topologically equivalent to)
2. Cross caps (All non-orientable closed surface topologically equivalent to)
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Mobius band - open surface w/ 1 edge ; non-orientable
Cross cap = sew mobius band into sphere w/ hole in it
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Fixed Points
Brouwer fix point theorem
X= topological space compact + convex
Compact = finite dim topo space bounded w/ all points of boundary
Convex = pt-to-pt enclosure (triangle,circle but not horshoe,torus)
f = continuous map of X to itself then f has fixed pt in X,
i.e. there exists x* in X s.t. f(x*)=x*
Sperner's Lemma
Chop up space
| ._. A/____|
| /A \ _____-- |
| / __\_- / | lim del X find fixed point
|------ \____/ B |
| |
+----------------------+
A A A B B A A
1. Cyclones - brouwer's fix pt -> can't comb wind to eliminate pt of zero vertical movement
2. Hohmann path - find map f whose fixed pts = solution optimal to newton eqn (hohmann path)
3. Equilibria class divisions - occupational mobility
4. Perron-Frobenius Theorem Ar=r Team score vector, pt of constancy
5. Adams (humanitraianistic economics)- L. Walrus (supply/Demand) - V. Neumann(equilib HI out Lo$, HI rate)
- Arrow (Gen equilib thy)
6. /\
/_p\ po->p1 in economic brouwer this is equilibrium pt
/p \ p1 s.t. apply T to p0 get to p1.
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