Question

With the standard topology on R,which one of the sets is open in R?

Answer 1

\[{x: \frac{1}{2}\ < |x| < 1} \]

Answer 2

i fink this is the answer but i have other options

Answer 3

so why is this open? Can you write it in interval notation?

Answer 4

(1/2,1)

Answer 5

but the absolute value sign means you also get (-1,-1/2). Do you see that?

1/2 < |-0.8|<1

Answer 6

ok,

Answer 7

So the set, in interval notation, is \[(-1,- \dfrac{1}{2})\cup (\dfrac{1}{2}, 1)\]

Answer 8

please why -0.8?

Answer 9

I was just showing you that you get negative number in there as well

Answer 10

\(\{x\mid \frac{1}{2}< |x|<1\}=(-1,-\frac{1}{2})\cup (\frac{1}{2}, 1)\)

Answer 11

ok. now i got it. because of the absolute value .

Answer 12

should i post the other options?

Answer 13

\[{x: \frac{1}{2} < |x|\leqslant1} \]

Answer 14

now this is neither open or close in R . right ?

Answer 15

Can you write it in interval notation?

Answer 16

please don't shout on me. i will try

Answer 17

what?

Answer 18

i mean please do not get angry if i fail

Answer 19

I don't even know how to respond to that.

Answer 20

what if i say it can not be written as union of open sets

Answer 21

(-1,-1/2)u(1/2,1]

Answer 22

you there?

Answer 23

almost, the -1 gets a [ also

Answer 24

i thought as much

Answer 25

\([-1, -0.4)\cup(0.5,1]\)

Answer 26

|-1|<=1

Answer 27

ok what is a closed set?

Answer 28

you gone?

Answer 29

a close set is a close interval. meaning that every set that has it boundary involve is close . because if we take a small radius around any boundary, it will not be in the set

Answer 30

no. i am here,network is getting bad

Answer 31

@zzr0ck3r

Answer 32

Ok, closed intervals are closed sets, but that is not all of them. R is both open and closed, and so is the empty set, and neither of those are closed intervals.

A set is closed if its compliment is open.

Do you know what a compliment is?

Answer 33

Ok, we can go with boundary, does this set contain all of its boundary points? If not, which 2 does it not contain?

Answer 34

Aight man, I am sorry about the internet, but this is taking way to long. It took about 35 minutes to get replies. I will be back tomorrow.

Answer 35

am sorry sir. my network is really poor

Answer 36

it does not contain all its boundary point. it contains only one which is 1 but does not contain 1/2. so it is half open and half close

Answer 37

Its boundary points are -1, -1/2, 1/2, and 1.

While it is not a great idea to think of every open/closed set as an interval, loosely said, the boundary points are the endpoints of any broken up union of sets.

Answer 38

so, x:1/2<|x|⩽1
is neither open nor close . so is it right to say that the only open sets in R in the options given above are \[x:1/2<|x|<1 \] and the close set in the option above in R is \[x:1/2\le|x|⩽1 \] right?

Answer 39

@zzr0ck3r

Answer 40

yes, unless I did not see all the options

Answer 41

ok sir

Answer 42

let me close this question and ask another . can i?

Answer 43

of course