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 xX and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)U.

Homotopy

Fibrations

Homotopy Lifting Property

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

Transport

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

EuclideanSpace

home

Where Are You

This is one of a series of pages about topology, these pages start here.

For other pages, on the site, about topology see:

• Open Sets
• Fibre bundles
• Simplectical Complexes
• Delta Complexes
• Homotopy
• Homology
• Cubical Type Theory

Sheaves and Category Theory

• Topoi.
• Presheaves
• Category of Presheaves
• Cubical Type Theory

Computer Code Implementation

I have written some code to implement algebraic topology structures. For an explanation of this code see the page here.

Prerequisites

• Topology
• Set theory explained on this page.