Download MA3056 — Exercise Sheet 2: Topological Spaces

30/1/04
MA3056 — Exercise Sheet 2: Topological Spaces
1. List all possible topologies on {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.
3. Let X be a topological space and A ⊂ X a subset. The boundary of A is defined to
be b(A) = A ∩ X \ A. Calculate b(A) when A = (0, 1] and
(i)∗ X = R, with the usual topology
(iii)∗ X = R, with the discrete topology
(ii) X = C, with the usual topology
(iv) X = C, with the indiscrete topology
4. Let {Tλ }λ∈Λ be S
a family of topologies on a set X. Show that
on X, but that λ∈Λ Tλ need not be.
T
λ∈Λ
Tλ is a topology
5.† Given a topological space (Y, S) and a map f : X → Y , show that the collection
Tf = {f −1 (U ) : U ∈ S} is a topology on X. Moreover, show that f is continuous with
respect to this topology, and that Tf is the weakest topology with this property.
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, and (iii) f (x) = x.
6.∗ 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.
7. Show that f : X → Y is continuous if and only if it is continuous as a map onto the
subspace f (X).
8.∗ 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.
9. 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 .
10. 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.
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 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 .
13. 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.
14. Prove that homeomorphism defines an equivalence relation on the class of all topological spaces.
15.∗ Show that if f : X → Y is a homeomorphism and A ⊂ X, then the restrictions
f |A : A → f (A) and f |X\A : X \ A → Y \ f (A) are both homeomorphisms.
16. 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 .
17. 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.
18. 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.
19. Show that in a Hausdorff space X, the set {x} is
(i) closed, and
(ii) the intersection of all open sets containing x.
20.∗ 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.
21. 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.
∗
†
Starred questions will be covered in tutorials
These are the assessed questions, and should be handed in by 5pm, 25/2/04