Download TOPOLOGY ASSIGNMENT 6 CONTINUOUS FUNCTIONS

TOPOLOGY ASSIGNMENT 6
CONTINUOUS FUNCTIONS, HOMEOMORPHISMS. PRODUCT SPACES.
FOR SUBMISSION: Q2-5, DUE NOVEMBER 18
IGOR WIGMAN
(1) Using your results in Assignment 5 Q4 decide for each of the functions of Assignment 5 Q4
whether it is a homeomorphism.
S
(2) Let {Ai }i∈I be a collection of subsets of a topological space X, such that X =
Ai . Let
i∈I
f : X → Y be a map of topological spaces such that the restriction of f to each Ai (equipped
with the subspace topology induced from X) is continuous.
(a) Show that if each Ai is closed, and I is finite, then f is continuous.
(b) Find an example where I is countable and each Ai is closed, but f is not continuous.
(c) Find an example where |I| = 2 (that is, X = A1 ∪A2 ), A1 is closed, but f is not continuous.
(3) (a) Prove or disprove: Any same cardinaly sets X, Y , equipped with the finite complement
topology Tf are homeomorphic. What about the countable complement topology Tc ?
(b) Let f : Rl → R be a function. Prove that f is continuous, if and only if it is right continuous
at all x ∈ R (i.e. lim f (x) = f (x0 )). Find a condition for f to be continuous as a funcx→x0 +
tion R → Rl , and Rl → Rl .
Q
(4) Let X =
Xi be a product of topological spaces, equipped with either the product or the
i∈I
box topology, A a topological space and f : A → X a function with coordinate functions given
by f (a) = (fi (a))i∈I .
(a) Prove that in this case, for both the product and the box topologies, if f is continuous
then all fi are continuous. (Hint: For this direction same proof as in HW5, Q6(d), works.)
(b) Find an example, where all fi are continuous, butQ
f , for X equipped with the box topology,
is not. (Hint: For example, consider Rω :=
R and the “diagonal” f : R → Rω ,
i∈N
f (t) = (t, t, . . .). Can you construct a set, open in the box topology, with pre-image {0}?)
Q
Xi be the
(5) Let {Xi }i∈I be a family of topological spaces with topologies TXi and X =
i∈I
product space.
(a) Suppose that for i ∈ I, Bi is a topology basis for the topology on Xi . Prove that
(
)
Y
B=
Ui : Ui ∈ Bi and Ui = Xi for i 6= i1 , . . . , in
i∈I
is a basis for the product topology on X, and B =
Q
Ui : Ui ∈ Bi
a basis for the box
i∈I
topology.
(b) Let {Xi }i∈I be a family of
Q topological spaces and Ai ⊆ Xi subspaces. Show that the
Ai coincides with the topology induced from X.
product topology on A =
i∈I
Q
(c) Prove that if for all i ∈ I, Xi is a Hausdorff space, then X =
Xi , equipped with the
i∈I
product topology, is Hausdorff. Using a the box topology being finer than the product
topology, deduce that the same holds for the box topology.
Q
(6) (a) Let I = [0, 1] and X =
R, i.e. an element of X is a function f : I → R.
i∈I
(i) Prove that a sequence {fn }n ⊆ X of real functions converges to some f ∈ X in the
product topology on X, if and only if it converges pointwise, i.e. for every x ∈ I,
fn (x) → f (x) in the usual sense of convergence of sequences.
(ii) Prove that if fn → f in the box topology, then fn → f uniformly? Construct an
example when fn → f uniformly, but not in the box topology. (Hint: for the second
task, think whether
fn ≡ n1 is convergent in the box topology).
Q
ω
(b) * Let X = R =
R, the collection of all real sequences. Let A ⊆ X be the set of
n∈N
“eventually zero” sequences, i.e. A = {{xn } : ∃N.∀n ≥ N. xn = 0}, and BN = {{xn } :
∀n ≥ N. xn = 0}. Find the closure A and BN for X equipped with either the product
topology or the box topology.
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