Download TOPOLOGY WEEK 3 Definition 0.1. A topological property is a

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.
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