Question

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.

Answer #1

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 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. )

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...

(i)LetX be a set andT,T′ two toplogies onX withT ⊂T′. What does
connectedness of X in one topology imply about connectedness in the
other?
2
(ii) Let X be an infinite set. Show that X is connected in the
cofinite topology.
(iii) Let X be an infinite set with the cocountable topology.
What can you say about the connectedness of X?

4. Which of the follows are vector spaces? Prove or
disprove.
(a) The set {x = αz | α ∈ R, z = (4, 6)T
}.
(b) The set of all 3 × 3 matrices which have all negative
elements

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

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)

Determine whether the following sets define vector spaces over
R:
(a) A={x∈R:x=k^2,k∈R}
(b) B={x∈R:x=k^2,k∈Z}
(c) C ={p∈P^2 :p=ax^2,a∈R}
(d) D={z∈C:|z|=1}
(e) E={z∈C:z=a+i,a∈R}
(f) F ={p∈P^2 : d (p)∈R}

Consider the following nonlinear system: x'(t) = x - y, y'(t) =
(x^2-4)y
a. Determine the equilibria.
b. Classify the equilibria using linearization.
c. Use the nullclines to draw the phase portrait.
Please write neatly. Thanks!

Determine which of the following equations are well defined
functions with an independent variable x. Explain.
A. 2y / x^2 - 3|x| = 1
B. Y^2 + x^2 = 10

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