Question

here

Answer 1

In the finite complement topology of R, let the sequence [x_{n} be defined by x_{n} = n, for n\epsilon N. If the limit of the sequence is x, then x must be

∞
0
A unique constant
Arbitrary in
R

Answer 2

@zzr0ck3r

Answer 3

what is the complement of R?

Answer 4

what is the set of all elements that are not in R

For the 6th times, please stop using epsilon. IT DOES NOT MEAN IN

Answer 5

i do not know

Answer 6

i thought there are two types of numbers

Answer 7

the imaginary and the real

Answer 8

when we talk about the real numbers as the universe, we do not consider them as a subset of the complex numbers.

The compliment of the real numbers is the emptyset

Answer 9

ok. thanks so much

Answer 10

so will that option b empty since is the complement of R?

Answer 11

In the cofinite topology,

1) a sequence that does not take an infinite many of the same value will converge to every element in the space.

2) A sequence that does(but only one number), will converge to that number

3) A sequence that has more than one number for which it obtains infinite times does not converge

Answer 12

So what is the answer?

Answer 13

Arbitrary in R

Answer 14

correct

Answer 15

i wonder how \(\{ x_n,\varnothing ,R
\}\) is considered cofinite :O

Answer 16

what does arbitrary really mean. because i see it as finite or infinite

Answer 17

um, it means if you close your eye and pick one

Answer 18

Arbitrary = any

Answer 19

@halmos? \(\{x_n, \emptyset, \mathbb{R}\}\) what is this?

Answer 20

In the finite complement topology of R, let the sequence [x_{n} be defined by x_{n} = n


was trying to interpret in notations

Answer 21

Arbitrary size means we are not given anything about the set, so as far as order, a set of arbitrary order is either finite or infinite. But we use the term this other way as well :)

Answer 22

he sucks with latex :)

Answer 23

he also usees \(\epsilon\) when he means \(\in\) so you got to be careful with that as well :)

Answer 24

oh i see now :)

Answer 25

Sorry @GIL.ojei I am just playing with you...

Answer 26

but seriously start using \(\in\) when you mean in. It is coded as `\(\in)`

Answer 27

\in

Answer 28

\(\epsilon\), in general, means a positive real number

Answer 29

@GIL.ojei u can use this next time for math notations :)
http://prntscr.com/8e7ekc

and yes epsilon used to denote very small values

Answer 30

thank u sir

Answer 31

or "arbitrarily small" positive number :)

Answer 32

ok

Answer 33

Let
(X,τ)
be a topological space, and let A be a subset of X. A is dense in X if and only if every non-empty open subset U of X, _________________

A⋂U=0

A⋃U=ϕ

A⋂U=ϕ

A⋂U=X

Answer 34

What is that weird symbol?

Answer 35

@GIL.ojei ?

Answer 36

this is just a definition, and i think the notation should be empty set u can use `\varnothing` \( \varnothing \) instead

Answer 37

empty set

Answer 38

then what is the 0?

Answer 39

i think he copied it from a place which typed it as \(\emptyset \) first (we see it 0) then as \(\phi\)

Answer 40

its just zero

Answer 41

that makes no sense

Answer 42

It should be intuitively clear from the definition of a dense set, that if \(U\) is open then \(A\cap U\ne \emptyset\).

But I do not see this option, and I dont know what the weird \(\phi \) looking thing is

Answer 43

or `\emptyset`

Answer 44

The union of any non empty set and anything else is non empty

Answer 45

these options don't make sense.

Answer 46

@GIL.ojei please reread the question and see if it shuold say \(\neq \emptyset\)

Answer 47

Let (X,τ) be a topological space, and let A be a subset of X. A is dense in X if and only if every non-empty open subset U of X, _________________

Answer 48

A⋂U=0

Answer 49

that is option a

Answer 50

A⋃U=\[ϕ \]

Answer 51

that is option b

Answer 52

A⋂U=\[ϕ \]

Answer 53

thats option C

Answer 54

A⋂U=X

Answer 55

option D

Answer 56

@zzr0ck3r

Answer 57

i think is option c

Answer 58

or option D

Answer 59

option a makes no sense

the union of subsets of some general topology are not necessarily equal to a number

option b is wrong because U is non empty

option c is wrong because of what I wrote about the non empty intersection

option d is only true sometimes

Answer 60

\(A\cap U \neq \emptyset\) is always true

Answer 61

Which of these is not true about
T1
- spaces?
a)Every singleton set is closed
b)Every finite set is closed
c)Every Hausdorff space is
T1

d)Y1isinR

Answer 62

this is the last question on topology for today. please, i will tell you what to teach me today . please

Answer 63

i know option A and B are correct

Answer 64

but what about C and D?

i fink C is also correct

Answer 65

what does Y1isinR mean and why do you keep writing things like this?

Answer 66

that was how i saw it

Answer 67

\[Y_1 \] is in R

Answer 68

you look at the page and it looks like this ?

Y1isinR

?

Answer 69

i think that was what they wanted to write

Answer 70

what is \(Y_1\)?

Answer 71

do not have idea but is Every Hausdorff space is T1?

Answer 72

Tell me the definitions of both

Answer 73

X is a T1 space if any two distinct points in X are separated.

Answer 74

i know that Hausdorff space has to do with intersection. but what is really separated?

Answer 75

what does separated mean?

Answer 76

i asked the question

Answer 77

So if it is Hausdorff, then for all \(x,y\) we have open set \(O,U\) such that \(x\in O, y\in U, O\cap U=\emptyset\)

Does this imply that there is a open set \(A\) that contains \(x\) and does not contain \(y\)?

Does this imply that there is a open set \(B\) that contains \(y\) and does not contain \(x\)?

IF the answer is yes, then we must be \(T_1\).

Answer 78

waw. so, T1 space is also Hausdorff space

Answer 79

we are not showing that T1 is Hausdorff, we are showing that Hausdorff is T1.

Answer 80

every horse has 4 legs but not every 4 legged animal is a horse

i.e. if a implies b it is not necessarily true that b implies a

Answer 81

ok

Answer 82

i want to close the tab and open another. i think there are thinks i want to understand