Space

Class Space

Overview

The Space class represents a topological space defined by a set of points and a set of open sets. A topological space in this library is a simplification, designed to work primarily with subsets of the real line.

Constructors

  • Space(std::set<double> points, std::set<std::set<double>> openSets): Creates a topological space given the set of points and the set of open sets representing the topology.

Member Functions

  • std::set<double> getPoints() const: Returns the set of points in the space.
  • std::set<std::set<double>> getOpenSets() const: Returns the set of open sets defining the topology.
  • bool isOpen(std::set<double> set): Determines if a set is an open set in the topology.
  • bool isNeighborhood(double x, std::set<double> set): Checks if a set is a neighborhood of the point x.
  • std::set<double> getComplement(std::set<double> set): Returns the complement of a set with respect to the space's points.
  • bool isClosed(std::set<double> set): Checks if a set is closed in the topology.

Example Usage

#include "homeomorphism.hpp"
 
// Create a topological space representing the interval (0, 1)
std::set<double> points = {0.1, 0.2, ... , 0.9};
std::set<std::set<double>> openSets = {{0.1, 0.2}, ... , {0.8, 0.9}};
topology::Space space(points, openSets);
 
// Check if a given set is open in this space
std::set<double> someSet = {0.2, 0.3};
bool isOpen = space.isOpen(someSet);
HomeomorphismSpace Property