M6140 Topology Exercises - 2nd Week (2020)
1 Subspaces
Exercise 1. Let A be a subspace of a topological space X. Prove that closed sets in A are precisely
the intersections of A with closed sets in X.
Exercise 2. Let A be a subspace of a topological space X and let B be a basis of X. Show that
{U ∩ A | U ∈ B} is a basis of A.
Exercise 3. Let X be a topological space. Prove that a subset U of X is open iff each point of U has
an open neighborhood V such that U ∩ V is open in V .
Exercise 4. Let A be a subspace of a topological space X and let ι: A → X be the inclusion. Show
that a mapping f : Y → A is continuous iff the composition ι ◦ f : Y → X is continuous.
2 Products
Exercise 5. Consider a product i∈I Xi of topological spaces. Prove that each projection map
pj : i∈I Xi → Xj is an open map, i.e. sends open sets to open sets.
Exercise 6. Consider a product i∈I Xi of topological spaces. Prove that if for each i ∈ I we have a
closed set Ci ⊆ Xi, then i∈I Ci is closed in i∈I Xi.
Exercise 7. Let A be a subspace of a topological space X and let B be a subspace of a topological
space Y . Show that the product topology on A×B coincides with the subspace topology when viewed
as a subspace of X × Y .
Exercise 8. Consider the product RN = N R of sets R. Give each component R the Euclidean
topology. Consider the diagonal mapping f : R → RN, f(x) = (x, x, x, . . . ).
(a) Prove that if RN has the product topology, then f is continuous.
(b) Prove that if RN has the box topology, then f is not continuous.
3 Disjoint Unions
Exercise 9. Show that a disjoint union of discrete spaces is discrete.
Exercise 10. Consider a disjoint union i∈I Xi of topological spaces. Show that each “injection”
ιj : Xj → i∈I Xi is both an open map and a closed map.
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4 Quotient Spaces
Exercise 11. Consider the relation on R given by x ∼ y iff x − y is an integer. Prove that this is an
equivalence relation and that the quotient space X/∼ is homeomorphic to the one-dimensional sphere.
Exercise 12. Let ∼ be an equivalence relation on a topological space X and let p: X → X/∼ be
the projection. Show that a mapping f : X/∼ → Y is continuous iff the composition f ◦ p: X → Y is
continuous.
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