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

Fibrations

Homotopy Lifting Property

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

Transport

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

 

 


metadata block
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
Correspondence about this page

Book Shop - Further reading .

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover Introduction to Topological Manifolds (Graduate Texts in Mathematics S.)

Other Books about Curves and Surfaces

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2021 Martin John Baker - All rights reserved - privacy policy.