Maths - Continuity

Definition 1

Here is one definition of continuity, based on open sets:

Let X and Y be topological spaces. A function f : X->Y is continuous if f-1(V) is open for every open set V in Y.

Definition 2

Another definition of continuity, based on neighbourhoods, is equivilant to the above definition.

Let X and Y be topological spaces. A function f : X->Y is continuous if for every x∈X and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)containsU.



Homotopy Lifting Property


from here
transport : Path U A B -> A -> B

That is, if we have a path from A to B and A, then B.

Composition of Paths

We want to compose these two paths:

  • p : Path A a b
  • q : Path A b c

from here

compPath (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : Path A a c =
   comp (<_> A) (p @ i)
                   [ (i = 0) ->  a
                   , (i = 1) -> q ]

Where 'comp' is a keyword with the following parameters:

  • a path in the universe.
  • the bottom of the cube we are computing.
  • a list of the sides of the cube



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see also:

Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory:

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
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