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arbitrary Assume base basic open set belongs bijection Cauchy sequence Choose closed interval closed sets closed subset collection compact space construct contains continuous functions continuous map converges Corollary countable d-ball d-bounded d-closed d-open sets deﬁned deﬁnition denote dense denumerable discrete space disjoint element equivalence class equivalence relation Euclidean metric example Exercise exists extended real line ﬁnite set ﬁrst ﬁrst-countable function f given Hausdorff space Hence homeomorphic inﬁnite intersection isometry Lemma Let f locally compact map f max metric metric space nonempty open interval open neighborhood open sets open subset positive integer product space product topology Proof Proposition Prove quotient map quotient space rational number real line real numbers relative topology satisﬁes second-countable subspace Suppose surjective T1-space Theorem topological property topological space uncountable upper bound usual topology