I know that boundary of union is subset of union of boundaries. When  does the equality hold? Can you give examples where there is equality and where there fails to be equality?
Assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$?   I can prove that $\partial (A \cup B)\subset \partial A\cup\partial B$ but for proving $\partial A\cup\partial B\subset \partial (A \cup B)$ it seems not trivial. I tried to show that for $x\in \partial A\cup\partial B$ WLOG, $x\in \partial A$ so $B(x)\cap A$ and $B(x)\cap A^c$ not equal to $\emptyset$ but it seems not enough to show the result.
Receiving Stack Overflow's Chat Notification This is regarding the feature where we can chat regarding a question on SO. There is an automated process for creating the room from comments. It's a very useful feature in case the OP is not able to understand an answer and needs more clarification.  What I don't see is a notification system for chats where there has been an extended period of time between communication.   Let's say someone replied to my chat after a few days, in that case I do not receive any notification.   This happened to me only a few hours ago, where in I had replied to one question but the OP was not available at that point and he replied to me after some days. As I didn't check that chat I had no way of noticing if someone replied to my message and the 48h window after my last activity in that chat, during which I would have received the @ convention ping notification, had already expired.  We receive notification on posts, upvotes that we receive, so why not the chat rooms?   Should there be a notification center like the one for posts (or may be include this in the same center as posts)? How to convert an array into comma separated strings in smarty template? How can I turn on the Intel HD manually? For some weeks my MacBook 6.2 does not switch to the integrated graphics adapter (Intel HD). This is especially crucial when I work on battery (which happens quite often). I guess the runtime could be enhanced when it would take the HD adapter for display.  I checked that it automatically changes the graphics mode. I am running Snow Leopard 10.6.8.  I recently installed 8GiB RAM (not from Apple). Could this probably related to this issue? Actually I do not want to downgrade or use the expensive Apple RAM.  So, how can I manually turn on the internal graphics adapter? Square root of Positive Definite Matrix Consider a common proof of the uncountability of $(0,1]$, as presented  for example:  We assume that the reals can be arranged in a sequence $x_k$, represent every number in $x_k$ by its nonterminating decimal expansion, use the diagonal argument to create a sequence of decimal digits that differs from the nonterminating decimal expansion of each number in $x_k$ and then claim that this sequence of digits so generated must be the decimal expansion of some number not in $x_k$.  My problem is with this last step. Could it not be the case that the new sequence of digits is just the terminating decimal expansion of some number already in $x_k$?  Would it not be more correct to use the sequence $x_k$ to generate a sequence of all (both terminating and non-terminating) decimal expansion of all numbers in $[0,1)$ and then apply the diagonal argument? Can I replace a 15 ampere circuit breaker with a 20 ampere breaker? When I plug-in a conventional heating pad or a hair dryer into any bedroom duplex, the lights dim; and/or trips the circuit breaker. The home is of new construction-- placed in service 31 Dec 2013. My question is: Can I safely replace the existing 15 AMP Breaker with that of a 20 AMP? . . . the service cable is of 14 Gauge Copper Conductor. Skyrim - Companions attacked me for no reason? Installing Elementary OS - will it automatically be added to the Grub menu? I currently have Windows 7 and Linux Mint installed.  When my PC starts up, it shows me the Grub menu, it shows an entry for Mint Linux and the Windows boot menu.  I am wondering, if I was to install Elementary OS, would it be automatically added to grub or do I have to put some command in the terminal?  I also want to ask, when I boot from my Elementary OS cd, and install it, just before it finishes, it shows some kind of error saying some error about cannot install to grub or something, I've tried installing to a separate partition and alongside "Them", but had no luck.  I'm really new to Linux, and want to keep my Mint Linux and Windows 7 because I have files on them. Why does Stack Overflow allow duplicate display names? What is the reasoning behind the design decision to allow users to create a display name which matches another existing users exactly?    This is opposed to the common practice of enforcing unique user names, and I'd really like to learn something. Proof of continuity of Thomae Function at irrationals. In :  $$ \begin{align} t(x) = \begin{cases} 0 &amp; \text{if $x$ is irrational}\\ \frac{1}{n} &amp; \text{if $x = \frac{m}{n}$ where $\gcd(m,n) = 1$} \end{cases} \end{align} $$  I can prove the discontinuity at rational $b$ by taking a sequence of irrationals $x_n$ which converge to $b$.   But while going through an argument for continuity at irrationals.  I found this in a book.      On the other hand if $b$ is an irrational number and $\epsilon &gt; 0$   then there is a     natural number $n_0$ such that $1/n_0 &lt; \epsilon$.   There are only finite number of rationals with denominator less than   $n_0$ in the interval $(b-1,b+1)$. Hence we can find a $\delta &gt; 0$   such that $\delta$ neighbourhood of $b$ contains no rational with   denominator less than $n_0$.   I understand the rest of the proof. But I am unable to prove the emphasized text. Although I find it intuitive. I've created main SELECT statement like below. For every row in Base table (base on IDKey I've added columns with values from result of INNER JOIN of few tables.  SELECT   IDKey,    Name,    (select Amount from Table1 INNER JOIN Table2 ... where ID=IDKey) AS AmountTypeX,    (select Amount from Table1 INNER JOIN Table2 ... where ID=IDKey) AS AmountTypeY FROM BaseTable   BaseTable:  IDKey Name  ---------- 1     Ian 2     John 3     Ann ...   INNER JOIN of few tables Table1, Table2... :   ID Amount Type -------------- 1  100    X 1  200    Y 2  300    X 2  400    Y 3  500    X 3  600    Y ...   Result looks like:  IDKey  Name  AmountTypeX  AmountTypeY ------------------------------------- 1      Ian   100          200 2      John  300          400 3      Ann   500          600 ...   Do you have any suggestion how to avoid this nested SELECT statements in the main query at the begining and made this query simpler and more effective ?  I wast thinking about PIVOTING of the INNER JOIN result to look like below (all data for particular ID in one row) and then JOIN with main query but Have no idea how to do this and is it possible.  ID  X   Y ----------- 1   100 200 2   300 400 3   500 600 ... Meaning of convolution? Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$ Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$.  Ok I'm finally understanding what each of these things mean.. but I'm running out of ideas on how to actually show it without using arbitrary vector space examples like Rn. Since sets and subspaces are kind of new its hard to figure out how to write correct proofs with them.   I can see how if $V = \mathbb{R}^3$ and $S = \{(1,1,1)\}$ and $W = \{(1,1,1), (1,1,2)\}$ then $\mathrm{span}(S) = R1(line), span(W) = R2(plane)$ and that R1 is a subspace of R2... im just having trouble showing this officially when all vector spaces and sets are completely in general....  I could get as far as writing out $S=(u_1,u_2,\dots,u_n)$, $u_i \in V$ and same with $W$, but I can't really figure out what else to do in this sort of proof. Infinite sum of ideals I've been trying to prove that given a ring $R$ and a collection of ideals $\{I_{\beta}\}_{\beta \in B}$ in $R$, the set $$\sum_{\beta \in B} I_{\beta}=\{f_1+\cdots+f_r:f_j \in I_{\beta_j}, \mbox{ for some } I_{\beta_j}\}$$ is an ideal and is the smallest ideal containing $\cup_{\beta \in B} I_{\beta}$.  I've already proved when I only have two ideals, and so, using induction, I've seen it for finite sums. But I don't see how to generalise that to an arbitrary family of indexes (maybe infinite) $\beta$.   Could someone please give me a hand? Thank you. How exactly do I install Minecraft Mods and what is Forge? I'm having a really, really hard time figuring out how to install mods on Minecraft. I've looked everywhere I can think of for help but the information is almost always outdated or unhelpful.  I've seen on several sites about the use of something called Forge. The Mod I want to use requires 1.6.4 Forge and I have both Forge and the Mod folders sitting in my downloads folder.  I haven't installed any kind of mods for any game prior to this and I have little clue to the technical side of Minecraft, so please keep this in mind when you give me an answer please!  Things that might help you figure out how to help me:   I want to play this mod  and am using the 1.6.4 Forge given on a link that website gives How do I set up SSH so I don't have to type my password? How do I set up SSH so I don't have to type my password when connecting to a host? Glassware with scratched/etched surfaces generate more bubbles (, ). Why these rough surfaces allow easier heterogeneous nucleation? Determining likely relationship based upon autosomal DNA shared cM How are app sales/acquisitions taxed in the US? Hypothetical scenario: I identify a gap in a company's product offering. I, as an independent developer, write a desktop, web, or mobile application that bridges this gap, and I contact the company in question. They like the product, and agree to acquire it from me.  To be clear, this scenario again isn't suggesting I've written an app and have put it on the App Store for continuous sale, but that I've written an app, and it has been acquired for a lump sum of money.  How is this taxed in the US? (Specifically, California.) Some thoughts I have:   Self-employment tax rate (currently 15.3%, I believe) CA sales tax (currently 7.25%) (Other income tax?)   Would this be taxed as a sale? Self-employment income? Both? What if I'm not self-employed, but the sale is much higher than my gross income? Example: $100,000 salary at company ___ that has no stake in the app invention, $1,000,000 app acquisition. Can I change my itinerary and hotel reservation after getting a Schengen visa?