Determine which of the following spaces are connected. (a). X=
{1,2,3,4} and T = {X, ∅,...
Determine which of the following spaces are connected. (a). X=
{1,2,3,4} and T = {X, ∅, {1}, {2, 3}, {1, 2, 3}}. (b). X = ℝ and T
= co-finite topology.
Define X = {0, 1} and T = {∅, {0}, X} .
(a) Is X with...
Define X = {0, 1} and T = {∅, {0}, X} .
(a) Is X with topology T connected? (Hint: Use the clopen
definition.)
(b) Is X with topology T path-connected? (Hint: Construct
continuous map f ∶ R → X. One way is to ensure f −1({0}) = (−∞, 0).
Once you have f, consider f([a, b])—like f([−1, 1]) if you take my
suggestion. Use the definition of path connected. )
For each of the following sets X and collections T of open
subsets decide whether the...
For each of the following sets X and collections T of open
subsets decide whether the pair X, T satisfies the axioms of a
topological space. If it does, determine the connected components
of X. If it is not a topological space then exhibit one axiom that
fails.
(a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3},
{1, 2, 3, 4}}.
(b) X = {1, 2, 3, 4} and T...
Let X be a topological space with topology T = P(X). Prove that
X is finite...
Let X be a topological space with topology T = P(X). Prove that
X is finite if and only if X is compact. (Note: You may assume you
proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2
and simply reference this. Hint: Ô⇒ follows from the fact that if X
is finite, T is also finite (why?). Therefore every open cover is
already finite. For the reverse direction, consider the
contrapositive. Suppose X...
Determine which of the following functions are injective
(one-to-one) on their respective domains and codomains
(a)...
Determine which of the following functions are injective
(one-to-one) on their respective domains and codomains
(a) f : ℝ → [0,∞), where f(x) = x²
(b) g : ℕ → ℕ, where g(x) = 3x − 2
(c) h : ℤ_7 → ℤ_7, where h(x) ≡ 5x + 2 (mod 7)
(d) p : ℕ ⋃ {0} → ℕ ⋃ {0}, where p(x) = x div 3
1)
x(t+2) = x(t+1) + x(t) , t >=0
determine a closed solution (i.e. a solution...
1)
x(t+2) = x(t+1) + x(t) , t >=0
determine a closed solution (i.e. a solution dependent only on
time t ) for above eqn. Verify your answer by evaluating your
solution at t = 0 , 1, 2, 3, 4, 5.
We are given x(0) = 1 and x(1) = 1
Determine how the following lines interact.
(x, y, z) = (-2, 1, 3) + t(1, -1,...
Determine how the following lines interact.
(x, y, z) = (-2, 1, 3) + t(1, -1, 5) ; (x, y, z) =
(-3, 0, 2) + s(-1, 2, -3)
(x, y, z) = (1, 2, 0) + t(1, 1, -1) ; (x, y, z) =
(3, 4, -1) + s(2, 2, -2)
x = 2 + t, y = -1 + 2t, z = -1 – t ; x = -1 - 2s,
y = -1 -1s, z = 1...
Let f(x)=(1/2)(x/5), x=1,2,3,4 Hint: Calculate F(X).
Find; (a) P(X=2) , (b) P(X≤3) , (c) P(X>2.5), (d)...
Let f(x)=(1/2)(x/5), x=1,2,3,4 Hint: Calculate F(X).
Find; (a) P(X=2) , (b) P(X≤3) , (c) P(X>2.5), (d) P(X≥1), (e)
mean and variance, (f) Graph F(x)