Topology
# Topology
Topology is a bad concept.
Open set is how open ?
Topology is not generalized concept.
It is unmatured and undeveloped concept.
How it will be if we can use (+) operator on Real Number,
but we cannot use it on Natural Number ?
Topology is so bad concept that it can not be built on Natural Number well.
# New definitions of topology
Grothendieck extract the instict structure of topology as the Grothendieck Topology $J$.
With a category consisting a poset $\mathscr{C}$ and $J$, we construct a Grothendieck Sheaf as a pair $(\mathscr{C}, J)$ which is equivalent to a presheaf $\mathscr{E}$ over $\mathscr{C}$.
There are three keywords to see contemporary topology.
- grothendieck topology
- frame
- locale
A locale is a topological space and a locale homomorphism is a continuous function of topological spaces.
A frame is a morphism in the category of opensets $\mathscr{O}_X$ over a topological space $X$.
A frame morphism is the pullback of a corresponding locale homomorphism.
Thus the category of locales $Loc$ and the cateogry of frames $Frm$ compose a duality.
# Presheaf Topos
What Grothendieck has made is a generalization of topological space.
We can handle discrete spaces ( simplicial complexes ) in the same way.
Presheaf over the category of simplex is called simplicial set $sSet$ and it is the very field of this new topology.
They can be glued and can be devided in a coherent way to discrete topological spaces.
What old topology could not do is, thus, generalised by geniouses, especially Grothendieck and Lurie.
# Glossary
## Sober
~~~
A topological space X is sober
if its points are exactly determined by its lattice of open subsets.
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## Lattice
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lattice : a poset which admits all finite meets and finite joins (or all finite products and finite coproducts,
regarding a poset as a category (a (0,1)-category)).
A lattice can also be defined as an algebraic structure, with the binary operations ∧
and ∨ and the constants ⊤ and ⊥.
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# Classic
## Morse Theory
Morse Theory := let $M$ be a manifold. Subeying around the singularities of $f : M \to \mathbb R$ determines the whole topology of $M$.