Find smallest $\sigma$-algebra so that it contains sets $\{x\}$, where $x \in \mathbb{R}$.  I was thinking of putting  $ F = \{ A\subset \mathbb{R} : A \text{ is countable or } A^{c}  \text{ is countable}\} $, but while I was proving it, I figured out that maybe that isn't it.
Let $S=\{\{x\}\mid x \in \mathbb{R}\}$. Is $\sigma(S)$ included in the Borel $\sigma$-algebra on $\mathbb{R}$? Is $\sigma(S)$ equivalent to the Borel $\sigma$-algbrea on $\mathbb{R}$?    I think the answer to the first question is yes. Because the Borel sets include singletons, the Borel $\sigma$-algebra must contain the smallest sigma-algebra of the set of all singletons.   Here's my attempt at the second question: my answer would be no. Countable unions (or intersections) of countable sets are countable, so that any $X \in \sigma(S)$ is either countable or a compliment of a countable set. Therefore, the interval $(0,1) \not \in \sigma (S)$ but it is included in the Borel $\sigma$-algebra. My issue here is that I don't know how to formalize this statement, if it is even correct.
How can I deploy a connected app from my sandbox environment to my production environment? I built a changeset and I was able to include all my classes, but it was a surprise that salesforce has not provided any option to deploy a connected app.  So how do I go about deploying a connected app? Do I need to create it in every other environment? I know that jQuery is loaded, because I can switch out the $ for 'jQuery' and everything behaves as expected, but this will be a messy script if I can't fix this  This script:  jQuery(document).ready(function(){     $("ul.vimeo_desc_feed li a").click(function(){         alert($(this).attr('href'));         return false;     }) });   Produces the error $ is not a function  This script:  jQuery(document).ready(function(){     jQuery("ul.vimeo_desc_feed li a").click(function(){         alert(jQuery(this).attr('href'));         return false;     }) });   works fine. I would really love to play wizard/warlock multiclass with low charisma (just 13 for mutliclass). The handbook says that I can use Pact Magic slots to cast wizard spells, but doesn't specify which ability I use for spellcasting.     Pact Magic.   If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and [vice versa].   When I cast wizard spells using Pact Magic slots (or warlock spells using wizard's "Spellcasting" slots) do I use spellcasting ability of the slot, or do I get to choose? Wrong nameserver set by resolvconf and NetworkManager How do I remove log out, reboot, and shut down from GNOME shell application list? I’m new to Ubuntu. I don't know where the log out, reboot and shut down icons in my application list came from. I just need to remove this from here. Comma usage with compound sentences {actually compound predicates as per Rob's answer} Rule 101 of compound sentences is that a coordinate conjunction must be accompanied by a comma most of the time when joining independent clauses, yet I stumble across sentences all that time that seem to abuse this rule. Normally, it's a case of the subject not being present in the second clause, as in the following example:      "They usually start out as an abbreviation, but turn into acronyms   when they become well known."   (taken from the Oxford Style Manual)   To me, this comma cannot be correct because the clause after the comma doesn't contain a subject and therefore cannot be independent (right?). As far as I'm concerned, it's not the 'Comma to set off a contrast rule because I can find plenty of other where 'and' is used and the subject's missing in the second clause.  what's the deal here?   Thanks in Advance for any responses I may receive! Solve congruence system $\ x\equiv m_i-1 \pmod{m_i}\,$ for $\,i = 1,\ldots, k$ The second part of an exercise asks that I show:     There does not exist a surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$.     The first part was to show that these two spaces are separable in their respective $L^p$ norms.    I've really hit a wall on this one.  I know, in fact, that there does exist a surjective bounded linear map from $\ell^1(\mathbf{N})$ to $\ell^2(\mathbf{N})$.  So my intuition says that if there were also one going the other way, then the two spaces would have to be isomorphic, which is false .  But I don't know of any such theorem that I could apply to "the category of Banach spaces".    My only other idea is to argue by contradiction using the open mapping theorem, but nothing is coming to me...  My guess is that separability is also going to be important, given the first part of the exercise.  -Thanks. Nowhere monotonic continuous function Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$? Some nowhere differentiable function sort of object? Questions with ignored tags still visible in iOS app I have set to ignore [php] tag and chose to hide questions in preferences:        But I still see questions tagged [php] in the iOS app.         Seems the app can't distinguish options for ignored tags.   App Version: 1.6.5.3 Device: iPad Air 2 (Wi-Fi) OS Version: Version 10.3.1 (Build 14E304) A virtual file containing the concatenation of other files Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? Mob Arena - Mob vs. Mob I want to make an arena where mobs will fight each other and a player can bet on one side or the other. For example: There are 2 skeletons in dyed armor; red and blue. Each color representing a team. When they spawn I want them to fight each and then when one dies I want it to announce which team is the winner and give the player the "money" due. I'm just not sure how to get the skeletons to attack each other without a player instigating it. Ubuntu touchpad issues - mouse pointer jumps around Moving from HTTP to HTTPS in Google Search Console Were any of the Elves in Tolkien twins? Thinking about the idea of Elves having low numbers overall, got me to wondering: as far as Tolkien's writing, could elves ever have twins?  As a "yes" answer, I need an example of twins (any of JRRT work). They must be full on Elves, not half-Men. As a "no" answer, has to be an explicit statement in his work stating Elves ALWAYS had one child at a time. Anything else is a "maybe, we don't know". Deduction of $\forall y \exists x (x=y)$ Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP Date Formatting C# Ubuntu 17.10 Keyboard not working after login following upgrade from 17.04 (Gnome version) I accidentally closed the window for the verfication code I wanted to create an account for Ubuntu One, but I got impatient when the email with the verification code didn't arrive... I thought I might have typed the wrong address, closed the window and tried again. it told me the address already existed and I just got the code. But since I closed the window I missed it. Is there any way to recover the code? I have a 64-bit machine that has Windows 7 32-bit and Ubuntu 32-bit installed as a dual boot setup. Can I run 64-bit software on top of these 32-bit OSes, or are they a "bottleneck" limiting the true capability of the underlying hardware?