Question

help

Answer 1

1 Every complete metric space is a ___________

Baire space

Blank space

Dense space

Cardinal space

Answer 2

Baire space :)

Answer 3

@Michele_Laino

Answer 4

i am here sir

Answer 5

ok!
a complete space is a space in which every Cauchy sequence converges in it

Answer 6

so, which means that Every complete metric space is a ___________

Baire space

Answer 7

don't get angry at me but what is a Cauchy sequence ?

Answer 8

a sequence is said a Cauchy sequence, if given \epsilon>0, there exists a natual number, say N, such thatfor each n, m two natural numbers, greater or equal to N, the subsequent condition holds:
\[\Large d_X\left( {{x_n},{x_m}} \right) < \varepsilon \]
where
\[\Large {d_X}\]
is the distance of your metric space X

Answer 9

now, every converget sequence is also a Cauchy sequence

Answer 10

ok

Answer 11

furthermore, if each Cauchy sequence converges to an element x of X, since X is complete

Answer 12

oops..I have made a typo, here is the right statement:
"each Cauchy sequence converges to an element x of X, since X is complete"

Answer 13

ok. thanks for that . now can we answer the questions i asked together sir?

Answer 14

i noticed it was a typo

Answer 15

since each convergent sequence is a Cauchy sequence, and each Cauchy sequence converges to an element of the space X it self, then X contains all its limits point, in other word, we have:
\[\Large \overline X = X\]
so X is a dense space

Answer 16

which means that a complete metric space is a dense space

Answer 17

yes!

Answer 18

thanks so much sir

Answer 19

ok! let's go to the next question, please

Answer 20

2 Let
(X,τ)
be a topological space. If
X
is second countable, then
X
is ____________countable

Third

Second

Fourth

First

Answer 21

i think first

Answer 22

If a topological space is second countable, then it is also first countable, sincce the second countability implies the first countability

Answer 23

since*

Answer 24

ok.

Answer 25

3 If
xϵA¯
, then there exists a sequence
(xn)
of A such that
xn→x
is only true if X is a(an) ___________________

Countable

Metrizable

Hausdorff

Separation

Answer 26

i think B

Answer 27

yes! I think so, since in order to speak about limit, we need of a topology, which can be induced by the metric of the space, so we need of a metrizable space

Answer 28

thanks.

Answer 29

7 Let
X=(a,b,c,d,e)andτ=(X,ϕ,[a],[c,d],[a,c,d],[b,c,d,e]).LetA=[a,c]]
, then set
A’ of limit points of A
is given by


A′=(b,c,e)

A′=(b,d,e)

A′=(b,e)

A′=X

Answer 30

please , i don't know things on that

Answer 31

i think (b,d,e)

Answer 32

x=a can not be a limit point

Answer 33

so, whatt should be the correct one

Answer 34

yes! I think so, it is {b,d,e}

Answer 35

please why is it (b,d,e)

Answer 36

please explain sir

Answer 37

since each neighborhood around x=b, d, e contains points of A other than b, d, e

Answer 38

ok sir

Answer 39

8 Let
R
, the real line be endowned with the discrete topology. Which of the following subsets of
R
is dense in
R


Q

Ritself

Qc
All singletons

Answer 40

here it is Q is dense in R, since we can show that between two real numbers, exists a rational number

Answer 41

i really need your help in this topology. i wish we can make out study time

Answer 42

9 Let
A=(0,1]⋃2
be a subset of
R
. Then the isolated points of
AinR
are

0 and1

0 and 2
1 and 2
[2]

Answer 43

x=0, 1 can not be isolated points

Answer 44

is it 0 and 2

Answer 45

and explain

Answer 46

no, x=0, is a limit point of A

Answer 47

so which?

Answer 48

[2] is a closed set in R, nevertheless it is an open subset of A, since it is given by the intersection between A and (-1, 4), so
I think [2]

Answer 49

hmm. so which are the limit points?

Answer 50

I think the answer is [2]

Answer 51

since I can find at least one neighborhood around x=2, such that it contains only x=2

Answer 52

ok thanks

Answer 53

10 For the set A in question above, which of the following are the limit points of
A
?

0 and1

0 and 2
1 and 2
2 only

Answer 54

x=0, and x=1

Answer 55

sir can you teach me four things here??

Answer 56

yes!

Answer 57

teach me the difference between usual real line and the standard real line

Answer 58

In general with the real line we indicate the set of the real number without the points:
\[ \Large + \infty ,\quad - \infty \]
furthermore, when we add those points to the real line, we get the so called "expanded line"

Answer 59

the so defined "expanded line" is again a totally ordered set

Answer 60

ok. please teach me the intersection and union of sets of real line. like

Answer 61

A=(0,1]⋃2

Answer 62

they are defined as usually. Namely, the intersection of two sets, is set of all points which belong to both those sets.
Similarly for the union of two sets, which is the set of the points which belong to one set or to the other set or to both
In your case, I think better is:
A=(0,1]⋃[2]

Answer 63

ok

Answer 64

do they mean (0,2)?

Answer 65

no, since the set (0,2) is:

Answer 66

x=0, and x=2 are not included

Answer 67

whereas the set A=(0,1] union [2], is: