Reference no: EM132641587

Question 1
Let X and Y be any topological spaces, and let f : X x Y → Y x X be the function defined by the formula
f (x × y) := y × x for all x × y ∈ X × Y.
(a) Is this function f continuous (at any or all points of its domain)?
(b) Is f bijective?
(c) If your answer is that f is bijective, then is f-1 continuous? Prove your assertions.

Question 2
Let X₁, X₂, Y₁, and Y₂ be any topological spaces. Let f₁ : X₁ -→ Y₁ and f₂ : X₂ → Y₂ be any functions. Now let f : X₁ x X₂ → Y₁ x Y₂ be the function defined by the formula
f (x₁ × x₂) := f₁ (x₁) × _f2 (x₂) for all x₁ × x₂ ∈ X₁ × X₂.
(a) If f₁ and f₂ are continuous, then is f continuous?
(b) If f is continuous, then are both (or either) of f₁ and f₂ also continuous? Prove your assertions.

Question 3

Let X, Y , and Z be any topological spaces, let f : X×Y -→ Z be a function, and let x₀ ∈ X and y₀ ∈ Y . Let g : X → Z and h : Y → Z be the functions defined by the formulas
g(x) := f (x × y₀) for all x ∈ X

and

h(y) := f (x₀ × y) for all y ∈ Y.

(a) If f is continuous, then are both (or either) of g and h continuous too? Prove your assertion(s).

(b) To show that the converse does not hold, consider the function
f : R × R -→ R defined by the formula

f (x × y) := xy/(x²+y²)   if x × y ≠ 0 × 0,

f (x × y) :=  0 if x × y = 0 × 0.

Let x₀ = y₀ = 0. Are the functions g and h constructed according to the above formulas for this particular function f continuous? Is this function f itself continuous? Prove your assertions.

Question 4

Let the set R of real numbers have the standard topology. Let f : R X R → R be the function defined by the formula

f (x × y) := x + y for all x × y ∈ R × R.

Is this function f continuous? Prove your assertion.

Question 5

Let the set R of real numbers have the standard topology. Let f : R X R → R be the function defined by the formula

f (x × y) := xy for all x × y ∈ R × R.

Is this function f continuous? Prove your assertion.

Question 6

Let the set R of real numbers have the standard topology. Let f : R → R be the function defined by the formula

f (x) := -x for all x ∈ R.

Is this function f continuous? Prove your assertion.

Question 7

Let the set R of real numbers have the standard topology. Let f : R x R → R be the function defined by the formula

f (x × y) := x - y for all x × y ∈ R × R.
Is this function f continuous? Prove your assertion.

Question 8

Let R have the standard topology, and let R\{0} have the subspace topology. Let the function f : R \ {0} -→ R be defined by the formula

f (x) := 1/x for all x ∈ R \ {0}.

Is this function f continuous? Prove your assertion.

Question 9

Let R have the standard t.opology, Σand let R\{0} have the subspace topology.

f (x × y) := x/y for all x × y ∈ R × R \ {0} .

Is this function f continuous? Prove your assertion.

Attachment:- Topology Assignment.rar