7. Let A be a subspace of a regular space X and consider the partition of X which consists of A and the singletons {x} for all x A. This partition defines an equivalence relation on X whose quotient space (with quotient topology) is called X/A. Prove that X/A is Hausdorff if and only if A is closed. 8. The second-countability is an essential part of the definition of a man- ifold: show that R2 with the dictionary order topology satisfies all the conditions for a 1-manifold except for second-countability. 7. Let A be a subspace of a regular space X and consider the partition of X which consists of A and the singletons {x} for all x A. This partition defines an equivalence relation on X whose quotient space (with quotient topology) is called X/A. Prove that X/A is Hausdorff if and only if A is closed. 8. The second-countability is an essential part of the definition of a man- ifold: show that R2 with the dictionary order topology satisfies all the conditions for a 1-manifold except for second-countability