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Problem set 2
Due in class on Monday, September 26.
1. Let B be the set of all closed intervals in R:
B = {[a, b] | a, b ∈ R}.
Show that B is a basis for a topology on R. If TB is the topology generated
by B, describe all of the open sets of B. Is TB coarser than, finer than,
equal to or incomparable with the usual topology on R?
2. Draw a rough picture of each of these subsets of R2 . Is it open, closed or
neither? Give proofs.
a)
{(x, y) ∈ R2 | 0 < x2 + y 2 < 1}.
b)
{(x, 0) ∈ R2 | x 6= 0}.
c)
{(x, αx) ∈ R2 | x ∈ R, α ∈ R>0 }.
d)
{(x, αx) ∈ R2 | x ∈ R, α ∈ R>0 } ∪ {(βx, x) ∈ R2 | x ∈ R, β ∈ R>0 }.
3. a) Endow R with its usual topology. Show that the sum and product
functions
f : R × R → R, f (x, y) = x + y
and
g : R × R → R, g(x, y) = xy
are continuous.
b) Let R be the topological space whose underlying set is R and whose
open sets are ∅, R and the rays
(a, ∞) = {r ∈ R | r > a}
for each a ∈ R. Are the sum and product functions
f : R × R → R, f (x, y) = x + y
1
and
g : R × R → R, g(x, y) = xy
continuous? Prove or disprove.
In this question it’s OK to be informal about whether a subset of R2 is
open.
4. A continuous map f : X → Y is called an open map if for every open set
U ⊆ X, its image f (U ) ⊆ Y is open. Similarly, f is called a closed map if
for every closed set Z ⊆ X, its image f (Z) ⊆ Y is closed.
Suppose K is a subset of X, equipped with its subspace topology, and let
i : K → X be the inclusion of K into X. Show that i is an open map if
and only if K is an open subset of X, and that i is a closed map if and
only if K is a closed subset of X.
5. Estimate how much time you spent on this problem set.
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