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Homework 3 Topology I, Fall 2014 Problem 1. (a) Show that a subspace of a completely regular space is completely regular. (b) Show that completely regular space is regular. (c) Show that the product of completely regular spaces is completely regular. Problem 2. Let X be a compact Hausdorff space. Show that X is metrizable if and only if it is second countable. Problem 3. Let X be a locally compact space and let A be a subset of X such that, for every compact subset K of X, the intersection A ∩ K is a closed subset of X. Show that A is a closed subset of X. Problem 4. Let X be a locally compact space and A a closed subspace of X. Show that A is locally compact. (Be careful, since we do not assume that X is Hausdorff, we cannot assume that compact subspaces are closed.) Problem 5. Show that a product of a paracompact space and a compact space is paracompact. Problem 6. Let X be a topological space and {Aj | j ∈ J} a locally finite family of sets in X. Show that {Aj | j ∈ J} is locally finite. Problem 7. Consider the space = R/ ∼, where x ∼ y ⇐⇒ x − y is rational. Show that X is an uncountable space with trivial topology. Problem 8. Consider the equivalence relation ∼ on I = [0, 1] given by x ∼ y ⇐⇒ x = y or 1/3 < x, y < 2/3, and the quotient space X = I/ ∼. Prove or disprove each of the following (a) X is Hausdorff. (b) X is connected. (c) X is compact.