Download Topology, MM8002/SF2721, Spring 2017. Exercise set 2 Exercise 1

Topology, MM8002/SF2721, Spring 2017. Exercise set 2
Exercise 1. Let f : X → Y be a map. Show that the following are equivalent:
• f is continous.
• f −1 (C) is closed for all closed sets C ⊆ Y.
• f −1 (A◦ ) ⊆ f −1 (A)◦ for all A ⊆ Y.
• f −1 (A) ⊆ f −1 (A) for all A ⊆ Y.
• The restriction f |U : U → Y is continous for all open subsets U ⊆ X.
Exercise 2. Let f : X → Y be a continous bijective map. Show that the following
are equivalent:
• f is open, i.e. images of open sets are open.
• f is closed, i.e. images of closed sets are closed.
• f is a homeomorphism.
• The restriction f |U : U → f (U ) is a homeomorphism for all open subsets
U ⊆ X.
Exercise 3. Let X be a topological space with the following property
• For every point p ∈ X there exists a continous function f : X → R such
that f −1 (0) = {p}.
Show that X is Hausdorff.
Exercise 4. Let X be a second countable space. Show that X contains a countable
dense subset.
Exercise 5. Consider the set X := (−∞, 0) ∪ {0, 00 } ∪ (0, ∞) = R ∪ {00 }.
• Show that {(a, b)| a < b ∈ R} ∪ {(a, 0) ∪ {00 } ∪ (0, b)| a < 0 < b ∈ R} forms
a basis for a topology.
• Show that the topological space given by the basis above is not Hausdorff.
• Show that every point in the topological space defined above has an open
neighborhood that is homeomorphic to an open set in R with it’s usual
metric topologty.
• Show that the topological space defined above is second countable.
Remark. The above is an example of a topological space that satisfies all the axioms
of a manifold besides being Hausdorff.
Can you give an example of a topological space that satisfies all the axioms of
a manifold besides being second countable? How about a second countable space
that is Hausdorff, but has a point that doesn’t have an open neighborhood that is
homeomorphic to an open subset of Rn ?
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