anonymous
  • anonymous
help
Mathematics
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
1 Every complete metric space is a ___________ Baire space Blank space Dense space Cardinal space
anonymous
  • anonymous
Baire space :)
anonymous
  • anonymous
@Michele_Laino

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anonymous
  • anonymous
i am here sir
Michele_Laino
  • Michele_Laino
ok! a complete space is a space in which every Cauchy sequence converges in it
anonymous
  • anonymous
so, which means that Every complete metric space is a ___________ Baire space
anonymous
  • anonymous
don't get angry at me but what is a Cauchy sequence ?
Michele_Laino
  • Michele_Laino
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
Michele_Laino
  • Michele_Laino
now, every converget sequence is also a Cauchy sequence
anonymous
  • anonymous
ok
Michele_Laino
  • Michele_Laino
furthermore, if each Cauchy sequence converges to an element x of X, since X is complete
Michele_Laino
  • Michele_Laino
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"
anonymous
  • anonymous
ok. thanks for that . now can we answer the questions i asked together sir?
anonymous
  • anonymous
i noticed it was a typo
Michele_Laino
  • Michele_Laino
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
anonymous
  • anonymous
which means that a complete metric space is a dense space
Michele_Laino
  • Michele_Laino
yes!
anonymous
  • anonymous
thanks so much sir
Michele_Laino
  • Michele_Laino
ok! let's go to the next question, please
anonymous
  • anonymous
2 Let (X,τ) be a topological space. If X is second countable, then X is ____________countable Third Second Fourth First
anonymous
  • anonymous
i think first
Michele_Laino
  • Michele_Laino
If a topological space is second countable, then it is also first countable, sincce the second countability implies the first countability
Michele_Laino
  • Michele_Laino
since*
anonymous
  • anonymous
ok.
anonymous
  • anonymous
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
anonymous
  • anonymous
i think B
Michele_Laino
  • Michele_Laino
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
anonymous
  • anonymous
thanks.
anonymous
  • anonymous
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
anonymous
  • anonymous
please , i don't know things on that
anonymous
  • anonymous
i think (b,d,e)
Michele_Laino
  • Michele_Laino
x=a can not be a limit point
anonymous
  • anonymous
so, whatt should be the correct one
Michele_Laino
  • Michele_Laino
yes! I think so, it is {b,d,e}
anonymous
  • anonymous
please why is it (b,d,e)
anonymous
  • anonymous
please explain sir
Michele_Laino
  • Michele_Laino
since each neighborhood around x=b, d, e contains points of A other than b, d, e
anonymous
  • anonymous
ok sir
anonymous
  • anonymous
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
Michele_Laino
  • Michele_Laino
here it is Q is dense in R, since we can show that between two real numbers, exists a rational number
anonymous
  • anonymous
i really need your help in this topology. i wish we can make out study time
anonymous
  • anonymous
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]
Michele_Laino
  • Michele_Laino
x=0, 1 can not be isolated points
anonymous
  • anonymous
is it 0 and 2
anonymous
  • anonymous
and explain
Michele_Laino
  • Michele_Laino
no, x=0, is a limit point of A
anonymous
  • anonymous
so which?
Michele_Laino
  • Michele_Laino
[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]
anonymous
  • anonymous
hmm. so which are the limit points?
Michele_Laino
  • Michele_Laino
I think the answer is [2]
Michele_Laino
  • Michele_Laino
since I can find at least one neighborhood around x=2, such that it contains only x=2
anonymous
  • anonymous
ok thanks
anonymous
  • anonymous
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
Michele_Laino
  • Michele_Laino
x=0, and x=1
anonymous
  • anonymous
sir can you teach me four things here??
Michele_Laino
  • Michele_Laino
yes!
anonymous
  • anonymous
teach me the difference between usual real line and the standard real line
Michele_Laino
  • Michele_Laino
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"
Michele_Laino
  • Michele_Laino
the so defined "expanded line" is again a totally ordered set
anonymous
  • anonymous
ok. please teach me the intersection and union of sets of real line. like
anonymous
  • anonymous
A=(0,1]⋃2
Michele_Laino
  • Michele_Laino
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]
anonymous
  • anonymous
ok
anonymous
  • anonymous
do they mean (0,2)?
Michele_Laino
  • Michele_Laino
no, since the set (0,2) is: |dw:1440443474555:dw|
Michele_Laino
  • Michele_Laino
x=0, and x=2 are not included
Michele_Laino
  • Michele_Laino
whereas the set A=(0,1] union [2], is: |dw:1440443551194:dw|
Michele_Laino
  • Michele_Laino
|dw:1440443643274:dw|
anonymous
  • anonymous
do what do they want us to do?
anonymous
  • anonymous
what do they want us to do?
Michele_Laino
  • Michele_Laino
they are representation of the two sets above
anonymous
  • anonymous
ok

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