Download TOPOLOGY WEEK 2 Definition 0.1. A topological space (X, τ) is

TOPOLOGY WEEK 2
Definition 0.1. A topological space (X, τ ) is second countable if there is a base B for τ such
that B is countable collection of sets. B is called a countable base.
(1) The usual topology on R denoted τR is generated by the Base, B = {(a, b) ∶ a < b}.
(a) Show that the usual topology on R is second countable. (Hint: Find countable
collection of sets A from B such that τ (A) = τ (B)).
(b) The Sorgenfrey Line topology on R is the topology generated by the base L =
{[a, b) ∶ a < b} . Show that L is a base and that τ (L) is strictly finer than τR .
(c) Prove or disprove: τ (L) is second countable.
(2) Let (N, τN ) be the one-point compactification of N. Let Y ∶= {0} ⋃{1/n ∶ n ∈ N, n ≠ 0}.
Show that Y with the subspace topology from the usual topology on R is homeomorphic to (N , τN ).
(3) The discrete topology on a space X is the topology of all subsets of X including ∅
and X. The indiscrete topology is the topology on X only containing ∅ and X. Prove
the following.
(a) If X has the discrete topology, then every function from X to a topological space
Y is continuous.
(b) If X does not have the discrete topology, then there is a topological space Y and
a function f ∶ X → Y that is not continuous.
(c) If Y has the indiscrete topology, then every function f from a topological space
X to Y is continuous.
(d) If Y does not have the indiscrete topology, then there is a topological space X
and a function f ∶ X → Y that is not continuous.
(e) Give an example of a continuous bijection f between two topological spaces such
that f is not a homeomorphism.
Date: April 1, 2015.
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