Homeomorphism

Class Homeomorphism

Overview

The Homeomorphism class is used to define a potential homeomorphism between two topological spaces and validate its properties.

Constructors

  • Homeomorphism(std::function<double(double)> function, std::function<double(double)> inverse, Space domain, Space range): Constructs a Homeomorphism given the mapping function, its inverse, and the two spaces being tested.

Member Functions

  • bool isValid(): Verifies if the constructed homeomorphism is valid by checking bijectivity, continuity, and the preservation of open and closed sets.
  • bool preservesOpenSets(std::function<double(double)> f, Space domain, Space range): Checks if a function preserves open sets.
  • bool preservesClosedSets(std::function<double(double)> f, Space domain, Space range): Checks if a function preserves closed sets.
  • bool isContinuous(std::function<double(double)> f, std::set<double> domain): Checks if a function is continuous.
  • bool isBijective(Space domain, Space range): Verifies if a function is bijective between two spaces.

Example Usage

#include "homeomorphism.hpp"
 
// Assume space1 and space2 have been defined (see SpaceClass.mdx for reference)
 
// Define the homeomorphism functions between space1 and space2
std::function<double(double)> homeoFunc = [](double x) { return 2 * x; };
std::function<double(double)> inverseFunc = [](double x) { return x / 2; };
 
// Create a Homeomorphism object
topology::Homeomorphism homeomorphism(homeoFunc, inverseFunc, space1, space2);
 
// Check if the homeomorphism is valid
bool isValidHomeo = homeomorphism.isValid();
Usage guideSpace