1
$\begingroup$

Let $ \left({{X}{\mathrm{,}}\mathit{\rho}}\right) $ be metric space . And the metric is the usual metric on X. where $ \hspace{0.33em}{X}\mathrm{{=}}\left[{{0}{\mathrm{,}}{3}{\mathrm{)}}\mathrm{\cup}\left[{{4}{\mathrm{,}}\left.{5}\right]}\right.}\right.\mathrm{\cup}{\mathrm{(}}{6}{\mathrm{,}}{7}{\mathrm{)}}\mathrm{\cup}\left\{{8}\right\} $

Then show if the following sets are open or closed .

a)(6,7)

b)(1,2)

C) $ \left\{{8}\right\} $

D) $ \left[{4\mathrm{,}5\mathrm{)}}\right. $

I tried by the complement to show that but I am a bit confused that the single point set is considered as a closed set in the example above it's complement isn't open .what can we say about this case . thanks in advanced ..

  • 4
    You should specify what the metric $\rho$ you are using in particular. This is from great relevance, because you can't answer the question if you don't know with which metric you are dealing with. – 2017-02-19
  • 0
    Ok sorry I forgot it. Can you see the question now . – 2017-02-19
  • 1
    Ok, I deleted my answer (peer pressure effect), maybe it was wrong, in that case I would have prefered someone to point out the fail in the reasoning, I'm also a student, I make mistakes and pointing them out is infinitely more useful for anyone (and in particular for me and for the OP) who read the post than simply bullying with the downvotes. If you are consideting X as a subspace of $\mathbb{R}$ with the induced metric (which is what I've assumed) then you can check the complement of $\{8\}$ is open because it is the intersection of X with an open ball in $\mathbb{R}$ – 2017-02-19
  • 0
    I just thought that it's neither open nor closed . – 2017-02-19
  • 0
    @user416990 no, it is both open and closed – 2017-02-19

2 Answers 2

0

HINT: I assume you are using the induced topology in $X$ from the standard topology on $\Bbb R$.

Observe that $(7.5,8.5)$ is open in $\Bbb R$ and $X\cap (7.5,8.5)=\{8\}$, hence $\{8\}$ is open in $X$. Can you continue from here?

  • 0
    No really why wiill we take the intersection I just was trying to find X/ $ \left\{{8}\right\} $ For example and then Judge if it's closed or open ... is that right ? – 2017-02-19
  • 0
    @user416990 but $\{8\}$ is both open and closed... observe that $X\cap(-1,7)=X\setminus\{8\}$ is open (and closed). – 2017-02-19
0
  1. Open: Let $x\in(6,7)$. Then $\exists r\in\Bbb R$ such that $B_r(x)\subset(6,7)$, for example you can choose $r=min\left\{\rho(x,6),\rho(x,7)\right\}$.
  2. Open: Same as 1.
  3. Closed and open: In fact $\mathcal C_X(\left\{8\right\})$ is open because it is union of open sets (can you do $[4,5]$ alone?). But also $\left\{8\right\}$ is open in $X$ because it is the intersection of $(8-\epsilon,8+\epsilon)$ with $X$, for some $\epsilon\in\Bbb R$ such that $0\lt\epsilon\lt1$.
  4. Open: $\forall x \in X$ there $\exists r\in\Bbb R$ such that $B_r(x)\subset [4,5)$. Take for example $r=min\left\{\rho(x,4),\rho(x,5)\right\}$