Download Topology, MM8002/SF2721, Spring 2017. Exercise set 3 Exercise 1

Topology, MM8002/SF2721, Spring 2017. Exercise set 3
Exercise 1. Consider the real numbers R with its standart topology. Show that
the function
(
0 if x ≤ 0
f : R → R, x 7→
1 if x > 0
is not continuous. Can you think of a (non discrete) topology on the domain, such
that f is continuous?
Exercise 2. Show:
• Every subspace of a Hausdorff space is Hausdorff.
• Every subspace of a first countable space is first countable.
• Every subspace of a second countable space is second countable.
Exercise 3. Let X be a topological space and let S be a subspace of X. Show
that the inclusion map S ,→ X is a topological embedding.
Exercise 4. Show that a surjective topological embedding is a homeomorphism.
Give an example of a topological embedding that is neither an open map nor a
closed map.
Exercise 5. Let B, B 0 be bases for the topologies of X and Y. Show that the set
{B × B 0 | where B ∈ B, B 0 ∈ B 0 } is a basis for the product topology on X × Y.
Exercise 6.
• Show that the inclusion map iy : X → X × Y, x 7→ (x, y) is a topological
embedding for any y ∈ Y.
• Show that the canonical projection πX : X × Y → X, (x.y) 7→ x is an open
map.
• Show that if X and Y are Hausdorff, so is X × Y .
• Show that if X and Y are first countable, so is X × Y .
• Show that if X and Y are second countable, so is X × Y .
Which of these statements hold for all finite products? Which for arbitrary
products?
Exercise 7. Consider an arbitrary product of topological spaces Πα∈A Xα . Show
that the two bases:
• {Πα∈A Uα | Uα is open in Xα }
• {Πα∈A Uα | Uα is open in Xα and Uα 6= Xα only for finitely many α}
induce non homeomorphic topologies. Note that the second one is a basis for
the product topology.
Exercise 8. Let X be a discrete topological space. Show
F that for any space Y the
product X × Y is homeomorphic to the disjoint union x∈X Y.
Exercise 9. Consider the set R2 . Let (a1 , b1 ), (a2 , b2 ) ∈ R2 , such that b1 ≤ b2 .
Consider the set B 0 given by the sets
• (a1 , a2 ) × {b1 } for b1 = b2 and a1 < a2
• (a1 , ∞) × {b1 } ∪ R × (b1 , b2 ) ∪ (−∞, a2 ) × {b2 } for b1 < b2 .
• Show that B 0 is a base for a topology on R2 . (It is called the lexicographic
order topology.)
• Show that R2 with the topology given by B 0 is homeomorphic to Rd × R,
where Rd denotes the set R with the discrete topology.
• Show that these spaces are not separable. Are they first countable?
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