2
$\begingroup$

Let $A_1$ and $A_2$ Two sets on the metric space $(X, \rho)$ show that :

a) $Cl(A_1\cup A_2) = CL(A_1)\cup Cl(A_2)$

b) $A_1\subset A_2 \implies Cl(A_1)\subset Cl(A_2) $

This is my attempt: $$ \text{If } x\in Cl(A_1\cup A_2)\\ B_\epsilon(x)\cap (A_1\cup A_2)\neq\varnothing\\ (B_\epsilon\cap A_1)\cup (B_\epsilon \cap A_2)\neq \varnothing\\ B_\epsilon\cap A_1 \neq \varnothing \implies x\in Cl(A_1)\\ \text{or }B_\epsilon\cap A_2 \neq \varnothing \implies x\in Cl(A_2) $$ For the second part: $$ \text{let } x\in A_1\implies x\in A_2\\ \text{if }x\in Cl(A_1)\implies B_\epsilon(x)\cap A_1\neq \varnothing\\ \text{but }A_1\subset A_2\\ \implies B_\epsilon(x)\cap A_2\neq\varnothing\implies x\in Cl(A_2) $$

  • 0
    This is true by replacing an $\epsilon$-neighborhood about a point to an arbitrary neighborhood, so the tags [tag:functional-analysis] and [tag:metric-spaces] aren't necessary. – 2017-02-19
  • 1
    You are aware that your math code can be written a _lot_ simpler? I don't think I've ever seen such dedication to $\LaTeX$ bloating. I'll see what I can tidy up, but it's gonna take a while. – 2017-02-19
  • 0
    Can you advice a program that I can use it for writhing ...and I will be very thankful. – 2017-02-19
  • 0
    You should use $\LaTeX$ (or more specifically MathJax) like you've done, but you've done so many unnecesary things to it. Click the edit button on the bottom of your question and see how I've written it. I'm not saying that it's perfect, but at least it's readable. Also, you could see [this guide](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) for general tips and tricks in the future. – 2017-02-19
  • 0
    Three pointers to your specific math code: Ease up on `\mathrm` and curly brackets. Curly brackets are (almost) only needed for when you want commands (like `\sqrt` or `^`) to take more than one symbol as input. `\mathrm` is mostly useful for specific text symbols that you don't want italicized, like the $\mathrm d$ in $\mathrm dx$. Also, using double dollar symbols instead of single dollar symbols and arrays is a lot easier and usually looks better for what it's for. Arrays are meant specifically to make tables. – 2017-02-19
  • 0
    @user416990 Both your answers are correct, and well-written as well. $+1$. I found out today that in order to close solved questions, the user should himself write out an answer, which he can accept after two days to complete the closure. I request you to do that. You may benefit from up votes, but my concern is entirely different – 2017-02-19

1 Answers 1

0

Your second part is ok, but the first is incomplete. You actually demonstrated that $Cl(A_1\cup A_2)\subset Cl(A_1)\cup Cl(A_2)$, but this doesn't mean they are equal. You should show also that $Cl(A_1)\cup Cl(A_2)\subset Cl(A_1\cup A_2)$.

  • 0
    Ok thanks a lot I got it – 2017-02-19
  • 0
    He actually did not show that inclusion (can you see why?hint: quantors), and the second one follows from his own b) applied twice. – 2017-02-19