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axioms bijection calculus Cauchy sequence Chapter claim of Example closed sets closed subset cluster point compact complement component connected subset connectedness continuous function converges Corollary defined Definition denote dense disjoint dominance elements equivalence relation EXERCISES FOR SECTION Figure final topology finite set finite subcovering fixed-point property formula function f geometry given Hausdorff property Hausdorff space hence homeomorphism identity integers interior operator left as exercises left for exercises Lemma Lindelof locally connected maps mathematics metric space nonempty set nonempty subset open covering open interval open sets open subset Outline of Proof pair path-connected point-set topology preceding theorem product topology proof is left proof of Theorem propositions Prove Theorem rational numbers real numbers relative topology satisfy second countable Section 16 segment separated subbasis subspace Suggestion for Proof Suppose surjection symbols topological space totally bounded triangle union usual topology Venn diagrams Verify the claim