Download List 6

MATH 701 - ELEMENTARY TOPOLOGY I
Exercise list 6
1. Let (X, TX ) and (Y, TY ) be nonempty topological spaces, and let A ⊆ X and B ⊆ Y be
proper subsets. If X and Y are connected, show that the complement of (A × B) is
connected.
2. Let (X, T) be a topological space, and let (Y, TY ) be a quotient space of X (that is,
Y = X/ ∼ for some equivalence relation ∼ on X , and TY is the quotient topology),
and let π : X → Y be the quotient map. Show that if Y is connected and π −1 ({y}) is
connected for all y ∈ Y , then X is connected.
3. Consider the following subset of R2 with the Euclidean topology:
Γ = {(x, 0) ∈ R2 | x ∈ [0, 1]}
[
{(1/n, y) ∈ R2 | n ≥ 1 and y ∈ [0, 1]}
[
{(0, 1)}.
Draw a sketch of Γ, and prove that it is connected but not path-connected.
4. Let (X, TX ) be a topological space, and let (R2 , TEucl ) be the Euclidean plane. Let also
S 1 = {(a, b) ∈ R2 | a2 + b2 = 1} ⊆ D2 = {(a, b) ∈ R2 | a2 + b2 ≤ 1}.
Let also T1 and T2 be the corresponding subspace topologies for S 1 and D 2 , respectively,
as subsets of R2 . Finally, let f : (S 1 , T1 ) → (X, TX ) be a continuous map, and let
Y = (X
a
D2 )/ ∼,
where (a, b) ∼ f (a, b) for each (a, b) ∈ S 1 , together with the quotient topology TY . Show
that if X is connected (respectively path-connected) then Y is connected (respectively
path-connected).
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