Let (X,7) be a topological space, and let ~ be a relation on X defined by...

  

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Let (X,7) be a topological space, and let ~ be a relation on X defined by ~ Y for , Y E X if there is a pathwise connected subset A of X such that , y E A Q. Discuss how you prove that is an equivalence relation on X. Once we prove this, we will have equivalence classes [x] = (y eX:y~x} in X with respect to . These equivalence classes are called path components of X X, which, therefore, form a partition of AX X into path connected subsets. Let (X,7) be a topological space, and let ~ be a relation on X defined by ~ Y for , Y E X if there is a pathwise connected subset A of X such that , y E A Q. Discuss how you prove that is an equivalence relation on X. Once we prove this, we will have equivalence classes [x] = (y eX:y~x} in X with respect to . These equivalence classes are called path components of X X, which, therefore, form a partition of AX X into path connected subsets.