Question

(a) This exercise will give an example of a connected
space which is

not locally connected. In the plane R2

, let X0 = [0, 1] × {0},

Y0 = {0} × [0, 1], and for each n ∈ N, let Yn = {1/n} × [0,1]. Let
Y = X0 ∪ (S∞

n=0 Yn). as a subspace of R

2 with its usual topology.

Prove that Y is connected but not locally connected. (Note
that

this example also shows that a subspace of a locally
connected

space need not be locally connected even if it is closed.)

Hint: There are many ways to prove this. One of them is as follows:
Define for each n ∈ N, Zn = X0 ∪ Yn and let Z0 = X0 ∪ Y0. State why
Zi is connected for each i ∈ N ∪ {0}. State why Y =S∞

i=0 union Zi is connected. To show that Y is

not locally connected, examine the point (0,1/2).

(b) Prove that an open subspace of a locally connected space is
locally

connected.

(c) Consider Y = {0} ∪ {1/n: n ∈ N} as a subspace of (R , U). Use Y
to show that a continuous image of a locally connected space need
not be locally connected even if the function is a bijection.

Hint: The set N ∪ {0} is discrete in (R , U). A similar argument
used in part

(a) could be used to show that Y is not locally connected.

Answer #1

Exercise 1.9.52. Give an example of a path connected space which is
not locally path connected

Topic: Calculus 3 / Differential Equation
Q1) Let (x0, y0,
z0) be a point on the curve C described by the following
equations
F1(x,y,z)=c1 , F2(x,y,z)=c2 .
Show that the vector [grad F1(x0,
y0, z0)] X [grad F2(x0, y0,
z0)] is tangent to C at (x0, y0,
z0)
Q2) (I've posted this question before but
nobody answered, so please do)
Find a vector tangent to the space circle
x2 + y2 + z2 = 1 , x + y +...

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

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Suppose that ? is a finite dimensional vector space over R. Show
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(please provide a detailed proof)
2. Suppose that ? is a finite dimensional vector space over R
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is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
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sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
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1. A function + : S × S → S for a set S is said to provide an
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all r, s, t ∈ S. Show that any associative binary operation + on a
set S can have at most one “unit” element, i.e. an element u ∈ S
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Uncorrelated and Gaussian does not imply independent unless
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Hint: use the deﬁnition of covariance cov[X,Y]=E [XY] −E [X] E [Y ]
and...

1. (a) Y1,Y2,...,Yn form a random sample from a probability
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Write the cumulative distribution function for Y(1) in terms of FY
(y) and hence show that the probability density function for Y(1)
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Let V be a finite dimensional vector space over R with an inner
product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
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