Usage guide

Usage Guide

Determining Homeomorphism

To determine if two topological spaces are homeomorphic, follow these steps:

  1. Create the Spaces: Define your topological spaces by specifying their points and open sets. See Space for more details on creating Space instances.

  2. Define the Homeomorphism: Define the bijective mapping functions that potentially create a homeomorphism between your spaces. You will need both the function and its inverse. The functions should be provided as std::function<double(double)> objects.

  3. Create a Homeomorphism Instance: Pass your spaces and the mapping functions to the Homeomorphism constructor.

  4. Validate the Homeomorphism: Call isValid() on the Homeomorphism instance to check if your mapping functions indeed establish a homeomorphism between the provided spaces.

Understanding the Results

  • If isValid() returns true, the spaces are homeomorphic.
  • If isValid() returns false, the spaces are not homeomorphic, or the functions provided do not establish a homeomorphism. You may need to recheck your spaces and functions for correctness.

Important Caveats

  • The library assumes the topological spaces are subsets of the real number line, \mathbb{R}, with standard topology.
  • All spaces are treated as Hausdorff spaces, as this is true for subsets of \mathbb{R}.
  • The isContinuous function uses a basic epsilon-delta definition of continuity and may not be suitable for complex spaces or functions.

Example: Verifying Homeomorphism

#include "homeomorphism.hpp"
 
// Step 1: Define the topological spaces
std::set<double> points1 = { /* set of points for space1 */ };
std::set<std::set<double>> openSets1 = { /* set of open sets for space1 */ };
std::set<double> points2 = { /* set of points for space2 */ };
std::set<std::set<double>> openSets2 = { /* set of open sets for space2 */ };
 
topology::Space space1(points1, openSets1);
topology::Space space2(points2, openSets2);
 
// Step 2: Define the homeomorphism functions
std::function<double(double)> f = [](double x) { /* define function */ };
std::function<double(double)> fInverse = [](double x) { /* define inverse function */ };
 
// Step 3: Create a Homeomorphism object
topology::Homeomorphism homeomorphism(f, fInverse, space1, space2);
 
// Step 4: Validate the homeomorphism
bool areHomeomorphic = homeomorphism.isValid();
 
if (areHomeomorphic) {
    std::cout << "The spaces are homeomorphic." << std::endl;
} else {
    std::cout << "The spaces are not homeomorphic." << std::endl;
}
Homeomorphism