Download MA3056: Exercise Sheet 2 — Topological Spaces

31/01/03
MA3056: Exercise Sheet 2 — Topological Spaces
1. List all possible topologies on (i) {a, b} (ii) {a, b, c}. Consequently find a
two topologies T1 and T2 on a set that are not comparable. That is, we have
neither T1 ⊂ T2 nor T2 ⊂ T1 .
2. Give an example of subsets A and B of R for which A ∩ B, A ∩ B, A ∩ B and
A ∩ B are all different.
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3. Let {Ti }i∈I be a family S
of topologies on a set X. Show that i∈I Ti is a
topology on X, but that i∈I Ti need not be.
Consequently, given a topological space (Y, S) and a map f : X → Y , show
that there is a weakest topology on X for which this map is continuous. Describe this topology when we take X = Y = R, S to be the usual topology,
and take f to be (i) a constant function, (ii) the function that maps (−∞, 0]
to 0 and (0, ∞) to 1 (iii) f (x) = x.
4. Prove that any map f : X → Y is continuous if either X is equipped with the
discrete topology or Y is equipped with the indiscrete topology.
5. Show that f : X → Y is continuous if and only if it is continuous as a map
onto the subspace f (X).
6. Give an example of topological spaces X and Y , a map f : X → Y and an
open set U ⊂ X such that f (U ) is not open in X.
7. Let (X, T) be a topological space and let U be an open subset of X. Show
that if V is a subset of U that is open in the subspace topology TU , then V is
open as a subset of X. Show that this can fail if we do not assume that U is
open.
8. Let X1 be a topological space, and X2 a subset of X1 equipped with the
subspace topology. Let A be a subset of X2 , and denote by Ai the closure of
A in Xi , for i = 1, 2. Prove that (i) A2 = X2 ∩ A1 (ii) if X2 is closed in X1
then A1 = A2 .
9. Let X1 , X2 , Y1 and Y2 be topological spaces, and let f1 : X1 → Y1 and
f2 : X2 → Y2 be maps. Define a map f : X1 × X2 → Y1 × Y2 by
f (x1 , x2 ) = f1 (x1 ), f2 (x2 ) .
Show that f is continuous if and only if f1 and f2 are continuous, where X1 ×X2
and Y1 × Y2 are given their respective product topologies.
10. Let X and Y be topological spaces and suppose that E ⊂ X and F ⊂ Y are
closed. Show that E × F is closed in the topological product X × Y .
11. (a) Let X1 and X2 be topological spaces, and let W be an open subset of
X1 × X2 . Show that pi (W ) is an open subset of Xi for i = 1, 2, where pi is the
projection map X1 × X2 → Xi .
(b) Give an example of a closed subset W ⊂ R × R such that p1 (W ) is not
closed in R.
12. Let X = [−1, 1] equipped with the usual topology.
(a) Let f : X → [0, 1] be the function f (x) = |x|. Show that quotient topology
induced on [0, 1] by f coincides with the usual topology.
(b) Find a surjection g : X → [0, 1] for which the quotient topology induced
by g is not Hausdorff.
13. Prove that homeomorphism defines an equivalence relation on the class of all
topological spaces.
14. Show that if f : X → Y is a homeomorphism and A ⊂ X, then g : A → f (A)
and h : X \ A → Y \ f (A) are both homeomorphisms.
15. Let X1 and X2 be topological spaces, and pick a2 ∈ X2 . Show that X1 is
homeomorphic to the subspace X1 × {a2 } of X1 × X2 respectively. Show that
X1 is homeomorphic to the subset {(x, x) : x ∈ X1 } of X1 × X1 .
16. Let X be an infinite set, and let T be the Zariski topology, that is,
T = {∅} ∪ {U ⊂ X : X \ U is finite}.
Show that T is indeed a topology, but that it is not Hausdorff.
17. Prove the following:
(i) Any subspace of a Hausdorff space is Hausdorff.
(ii) The product of two Hausdorff spaces is Hausdorff.
(iii) If f : X → Y is continuous and injective, and Y Hausdorff, then so is X.
(iv) Being Hausdorff is a topological property, that is, it is preserved by homeomorphisms.
18. Show that in a Hausdorff space X, the set {x} is (i) closed, (ii) the intersection
of all open sets containing x.
19. Let f, g : X → Y be continuous maps between topological spaces, with Y
Hausdorff. Show that W = {x ∈ X : f (x) = g(x)} is closed in X. Deduce
that if f : X → X is a continuous map and X is Hausdorff then the fixed
point set {x ∈ X : f (x) = x} is closed.
20. A T1 -space is a topological space (X, T) that satisfies the following: for any
pair of distinct points x, y ∈ X there are Ux , Uy ∈ T such that
x ∈ Ux , y ∈ Uy , but x ∈
/ Uy , y ∈
/ Ux .
Show that every Hausdorff space is a T1 -space, but that there are T1 -spaces
that are not Hausdorff.