Point Set Topology

Munkres

Leinster - General topology bourbaki

Topology - A space equipped with a set of open sets. https://en.wikipedia.org/wiki/Topological_space

-- fiddling
structure Topology (A : Type) where
open_sets : Set (Set A)
empty_open : (fun x => False) ∈ open_sets
full_open : (fun x => True) ∈ open_sets
union_open :
inter_open : forall x y : Set a, open_sets x -> open_sets y -> open_sets (fun z => x z /\ y z)



https://gist.github.com/andrejbauer/d31e9666d5f950dd8ccd andrej bauer coq. syntehtic topology ad functions into sierpinski space https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space https://github.com/coq-community/topology

https://en.wikipedia.org/wiki/Filter_(mathematics) many presentation use filters https://en.wikipedia.org/wiki/Filters_in_topology

https://en.wikipedia.org/wiki/Axiomatic_foundations_of_topological_spaces closure and interior operators can be axiomatized

Continuity

Continuous functions have open preimages of open sets.




quotient topology. Take an equivalence relation. The quotient set is the set of equivalence classes [z] = {x | x ~ z} The quotient toplogy is the toplogy whose

Homeomorphism

https://en.wikipedia.org/wiki/Triangulation_(topology)

Brouwer fixed point https://en.wikipedia.org/wiki/Sperner%27s_lemma

jordan curve theorem

covering spaces

fundamental polyhedron - polyhedra schema. Those diagram with glued edges. The edges are generators

Algebraic Topology

algerbaic topology hatcher

Peter May - Algebraic Topology a concise course

Complexes

# toy around with hap definitions

class CW():

#triangle
points = [None,None,None]
edges = [(0,1), (1,2), (2,0)]
faces = [(0,1,2)] # indices refer to previous

# square
points = [None,None,None,None]
edges = [(0,1), (1,2), (2,3), (3,0)]
faces = [(0,1,2,3)] # indices refer to previous


Discrete finite representation of CW uses loops over previous layers. The “disk” is a polygon. This is nice compared to simplicial because there is a nicer notion of “simplifying” a space by removing redundant subdivision.

Regular vs non regular

Homotopy

Deformation Retract- $f_t : X -> X$ s.t. $f_0 = id$ and f_1 = A and f_t(A) = A for all t. It leaves A alone for all t. https://en.wikipedia.org/wiki/Retraction_(topology). Retraction is projection to subspace that preserves all points in subspace.

https://en.wikipedia.org/wiki/Inclusion_map

Computational Topology

https://en.wikipedia.org/wiki/Computational_topology https://www.computop.org/ lots of cool programs

haskell rewrite of kenzo

http://snappy.computop.org/ snappea for 3 manifolds. python bindings http://snappy.computop.org/verify_canon.html canonical retriangulation / cell decoomposition geometrization theorem dehn-filling There’s like a canonical geometrical surface? And then metrical things like volume are cannical?

uniformizaton

http://chomp.rutgers.edu/ chomp computational homology

exact reals? hott?

toploogical data anaysis persistent topology

seifert

computational group theory MAF https://maffsa.sourceforge.net/manpages/MAF.html KBMAG automata to describe the multiplication operation

Resources

https://news.ycombinator.com/item?id=39396337
Data Structures as Topological Spaces (2002) [pdf] (spatial-computing.org) http://mgs.spatial-computing.org/