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anonymous
 one year ago
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anonymous
 one year ago
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anonymous
 one year ago
Best ResponseYou've already chosen the best response.01 Every complete metric space is a ___________ Baire space Blank space Dense space Cardinal space

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0ok! a complete space is a space in which every Cauchy sequence converges in it

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, which means that Every complete metric space is a ___________ Baire space

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0don't get angry at me but what is a Cauchy sequence ?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0a 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
 one year ago
Best ResponseYou've already chosen the best response.0now, every converget sequence is also a Cauchy sequence

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0furthermore, if each Cauchy sequence converges to an element x of X, since X is complete

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0oops..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
 one year ago
Best ResponseYou've already chosen the best response.0ok. thanks for that . now can we answer the questions i asked together sir?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i noticed it was a typo

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0since 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
 one year ago
Best ResponseYou've already chosen the best response.0which means that a complete metric space is a dense space

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0ok! let's go to the next question, please

anonymous
 one year ago
Best ResponseYou've already chosen the best response.02 Let (X,τ) be a topological space. If X is second countable, then X is ____________countable Third Second Fourth First

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0If a topological space is second countable, then it is also first countable, sincce the second countability implies the first countability

anonymous
 one year ago
Best ResponseYou've already chosen the best response.03 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

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0yes! 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
 one year ago
Best ResponseYou've already chosen the best response.07 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
 one year ago
Best ResponseYou've already chosen the best response.0please , i don't know things on that

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0x=a can not be a limit point

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, whatt should be the correct one

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0yes! I think so, it is {b,d,e}

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0please why is it (b,d,e)

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0since each neighborhood around x=b, d, e contains points of A other than b, d, e

anonymous
 one year ago
Best ResponseYou've already chosen the best response.08 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
 one year ago
Best ResponseYou've already chosen the best response.0here it is Q is dense in R, since we can show that between two real numbers, exists a rational number

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i really need your help in this topology. i wish we can make out study time

anonymous
 one year ago
Best ResponseYou've already chosen the best response.09 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
 one year ago
Best ResponseYou've already chosen the best response.0x=0, 1 can not be isolated points

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0no, x=0, is a limit point of A

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0[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
 one year ago
Best ResponseYou've already chosen the best response.0hmm. so which are the limit points?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0I think the answer is [2]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0since I can find at least one neighborhood around x=2, such that it contains only x=2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.010 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sir can you teach me four things here??

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0teach me the difference between usual real line and the standard real line

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0In 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
 one year ago
Best ResponseYou've already chosen the best response.0the so defined "expanded line" is again a totally ordered set

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok. please teach me the intersection and union of sets of real line. like

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0they 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]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0no, since the set (0,2) is: dw:1440443474555:dw

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0x=0, and x=2 are not included

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0whereas the set A=(0,1] union [2], is: dw:1440443551194:dw

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0dw:1440443643274:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0do what do they want us to do?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what do they want us to do?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.0they are representation of the two sets above
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