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TOPOLOGY WEEK 3 Definition 0.1. A topological property is a property of a topological space that is preserved under homeomorphisms. P is not a topological property if there exists two homeomorphic topological space for which one space has P, while the other spaces does not have P. Definition 0.2. A topological space is metrizable if it is homeomorphic to a metric space. (1) Prove that given any set X, the discrete topology on X is metrizable. (2) Prove that any open interval of R with the usual topology is homeomorphic to R with the usual topology. Is completeness a topological property? Why or why not? (3) Prove that separability is a topological property. (4) Prove that second countability is a topological property. (5) The Sorgenfrey Line topology on R is the topology generated by the base L = {[a, b) ∶ a < b} . (a) Prove that (R, τ (L)) is separable. (b) Prove or disprove: (R, τ (L)) is metrizable. (Hint: What can be said about separable metric spaces in relation to second countable?). Date: April 8, 2015. 1