1. The problem statement, all variables and given/known data Consider a topological space X Show that every point of X is contained in a unique path component, which can be defined as the largest path connected subset of X containing this point. 3. The attempt at a solution What happens if we take X=Q? There are no path connected subsets of Q. Or would in this case the path components are the sets containing the individual elements of Q? Which form the basis for Q. So since there exists a basis for every topological space, the elements of the basis of X are always path components.