here

- anonymous

here

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- anonymous

Let R be endowed with the usual standard topology. Consider Y = [-1,1] as a subspace of R. Which one of the following sets is closed in Y

- anonymous

@zzr0ck3r

- zzr0ck3r

?

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## More answers

- anonymous

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

- zzr0ck3r

not closed

- anonymous

is that open?

- anonymous

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

- zzr0ck3r

yes, since it is open in R and and it equals the intersection with the subspace.

- zzr0ck3r

its open

- anonymous

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

- zzr0ck3r

neither open nor closed

- anonymous

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

- zzr0ck3r

thats the same. please use parentheses. bbl

- anonymous

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

- anonymous

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

- anonymous

i presume this is close

- zzr0ck3r

yep

- zzr0ck3r

closed sets in a subspace S, are of the form \(A\cap S\) where \(A\) is open in the parent topology.

- zzr0ck3r

ok walking out the door

- anonymous

and which is open in Y with same options ?

- anonymous

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

- anonymous

2 With the standard topology on R,which one of the sets in question (1) above is open in R? with same options given above?

- anonymous

With the standard topology on R,which one of the sets in question (1) above is closed in R? with same options given above

- zzr0ck3r

hi

- anonymous

sir, i am here @zzr0ck3r

- zzr0ck3r

ok

- anonymous

so, can you help with the asked question sir?

- zzr0ck3r

\((-1, -\dfrac{1}{2})\cup (\dfrac{1}{2}, 1)\) is open
\((-1, -\dfrac{1}{2}]\cup [\dfrac{1}{2}, 1)\) is not open
\([-1, -\dfrac{1}{2})\cup (\dfrac{1}{2}, 1]\) is not open
\([-1, -\dfrac{1}{2}]\cup [\dfrac{1}{2}, 1]\) is not open

- anonymous

so which means that \[ x:1/2 <|x|<1 \] is open in the usual standard toplogy

- zzr0ck3r

correct

- anonymous

what about in the standard topology on R

- anonymous

2 With the standard topology on R,which one of the sets in question (1) above is open in R with same options given above?

- anonymous

you here sir?

- anonymous

@zzr0ck3r

- anonymous

hello sir, @zzr0ck3r

- anonymous

let me post the question in full . please don't get mad at me

- anonymous

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

- anonymous

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

- anonymous

is the answer still the first here sir?

- anonymous

i guess from what you thought, A is the only open set while B,C, D are close sets

- anonymous

am i correct?

- anonymous

another question

- anonymous

A topological space is called a Hausdorff space, if for each x, y of distinct points of X, there exists nbds \[U_{x} \] and \[U_{y} \] of x and y respectively that are disjoint. This implies X is Hausdorff wiith these properties except one.

- anonymous

A) if \[ \forall x, y \epsilon\mathbb X; x \neg y \]

- anonymous

B)\[There \exists U_x \epsilon N(x). U_y \epsilon N(y) \]

- anonymous

C) \[U_x \bigcap U-y = \phi \]

- anonymous

D)\[\forall x, y \epsilon X, x \bigcap y = 0 \]

- zzr0ck3r

what is the question?

- anonymous

should i post it in anew question?

- zzr0ck3r

yes

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