In real analysis class, the professor used a basic property to give an easier definition of measurable function, it says:  Any open set $G\subset\mathbb{R}$ has the form $G=\sqcup_i (a_i,b_i)$.  It is an axiom? or how to prove it?   By the way, it seems I need some background knowledge of this kind of set properties. Are there some materials I can refer?
Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals.    This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as possible. A professor told me that there are many. So, I invite everyone who has seen proofs of this fact to share them with the community. I think it is a result worth knowing how to prove in many different ways and having a post that combines as many of them as possible will, no doubt, be quite useful. After two days, I will place a bounty on this question to attract as many people as possible. Of course, any comments, corrections, suggestions, links to papers/notes etc. are more than welcome.
How to make a VGA monitor display a white screen at all time? How can I stream live video on YouTube? Nowadays, I see a lot of users live streaming on YouTube and wanted to try it for myself. However, when I go to my channel I don't see an option for live streaming.  I tried researching how to do this on YouTube and Google, but without luck.  So I was wondering, how can I live stream on YouTube?  I don't have any pro account or any of that, but I've watched livestreams from users who I don't think have pro accounts either.  So livestreaming is open for everyone, right? Can the manual and tome magic items that increase stats be used multiple times by bypassing the century wait time via the spell Sequester? Assume we have a Wizard who is at least level 13 and can reliably cast  many times (either through acquired spell scrolls, or has acquired enough material components for it to be a non-issue), has a safe place (through the spell ) to be under the influence of Sequester for many centuries, and has access to the Manual of Bodily Health, Manual of Gainful Exercise, Manual of Quickness of Action, Tome of Clear Thought, Tome of Leadership and Influence, and Tome of Understanding.  Per the item descriptions in the DMG:     ...your [STAT] score increases by 2, as does your maximum for that score. The manual then loses its magic, but regains it in a century.   Now, normally this would be a once-in-a-lifetime use, or maybe twice (due to the lifespan of the races), but if a wizard used Sequester on themself, many years can pass without growing older:     ...[willing target creature] falls into a state of suspended animation. Time ceases to flow for it, and it doesn't grow older. You can set a condition for the spell to end [before the spell is dispelled by the caster].   So, assuming that time continues to flow for the magic items, but not for the sequestered wizard, can they study these books and take the necessary long rests, Sequester for a hundred years,  “wake” from the Sequester, and then repeat an arbitrary-but-finite number of times for an arbitrarily high (but finite) improvement to their stats?  To end the cycle and escape the demiplane, the wizard would simply Plane Shift out:     ...You can specify a target destination in general terms ... and you appear in or near that destination. Noticing that Newtonian gravity and electrostatics are equivalent, is there also a relationship between the general relativity and electrodynamics? In classical mechanics, we had Newton's law of gravity $F \propto \frac{Mm}{r^2}$.  Because of this, all laws of classical electrostatics applied to classical gravity if we assumed that all charges attracted each other due to Coulomb's law being analogous.  We can "tweak" classical electrostatics to fit gravity.  In modern physics, does the reverse work?  Can we "tweak" General Relativity to accurately describe electrostatics or even electromagnetism? How can a distribution have infinite mean and variance? It would be appreciated if the following examples could be given:   A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with finite mean and infinite variance.  A distribution with finite mean and finite variance.   It comes from me seeing these unfamiliar terms (infinite mean, infinite variance) used in an article I am reading, googling and reading a thread on the , and not finding it a sufficiently clear explanation. I also haven't found any explanations in any of my own textbooks. Why don't we define division by zero as an arbritrary constant such as $j$? Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is the only number such that $0*j \not= 0$ , and put other restrictions necesary so that we don't get contradictions. We do this for $i$ , so why can't we do it here? For example, $i^2$ , is defined as $-1$ , but you could also say $i^2=\sqrt {-1} *\sqrt {-1}=\sqrt {(-1)(-1)}=\sqrt{1}=1$ , but we make an exception for this. Precedence between implication and bi-implication I came across this question:  Let p, q, and r be the propositions  p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (including negations).     If berries are ripe along the trail, hiking is safe if and   only if grizzly bears have not been seen in the area.   I see an If r,q iff ~p where If r,q is the part "If berries are ripe along the trail, hiking is safe" and ~p is "grizzly bears have not been seen in the area"  and I came up with (r -&gt; q) &lt;-&gt; ~p  But the answer given is: r -&gt; (q &lt;-&gt; ~p)  Can someone please explain the difference in precedence? In plain English, how does casting spells and using creature abilities work with the stack and priority? Change date format [YYYYMMDD] to [MM/DD/YYYY] Why do you write "occurred" but "listened"? The past tense of to occur is occurred (not occured), but the past tense of to listen is listened (not listenned).  Why? What is the general rule that is applied to make the past tense of a verb? Suppose I have a program named any_program.exe and my operating system drive is C:. The location of the program is D:\Any_Folder\any_program.exe  How do I start/execute that program via command prompt in Windows 8?  I have tried the command line START any_program.exe, but it shows me an error that      Windows cannot find 'any_program.exe'. Make sure you typed the name correctly, and then try again.   By the way, it worked perfectly in Windows 7. And, if I type START notepad.exe or START firefox.exe (Firefox is not installed in C: drive), it works in Windows 8. Let $C$ be a finitely generated $R$-algebra where $R$ is unital commutative ring. Let $S$ be a simple $C$-module. Show that the annihilator of $S$ in $R$ is a maximal ideal. What I have tried is: When $C=R$ then it is a direct consequence of the simplicity of $S$ as an $R$-module. Further, since $S$ is simple then $S$ is cyclic (i.e. $S=Cy$ for nonzero $y\in S$). Consider the map $$f: C \to S$$ $$c \mapsto cy$$ Then $\ker(f)$ is the annihilator of $S$ in $C$ and $\ker(f) \cap R$ is the annihilator of $S$ in $R$. I have shown that it is an ideal of $R$ but I have no idea to continue showing the maximality. Is it a true direction or is there anything else I should consider to show the maximality? #1115 - Unknown character set: 'utf8mb4' When I Import Mysql from local server to web ser it is showng me error 1115 - Unknown character set: 'utf8mb4' in Creating wordpress wp_comment table CREATE TABLE IF NOT EXISTS  `wp_commentmeta` ( // // ) ENGINE = INNODB DEFAULT CHARSET = utf8mb4 AUTO_INCREMENT =1;  #1115 - Unknown character set: 'utf8mb4'  My mysql version is Version information: 4.0.4, latest stable version: 4.4.10 (local version)  Version information: 4.0.10.7, latest stable version: 4.4.10( server version)  Please help me to solve out the error How do I cd into a directory in the home folder? I'm using Python, and would like to insert a string into a text file without deleting or copying the file. How can I do that? How do I install the Intel Graphics driver in my system? Can someone help me out and explain or point me in the right direction on how to check video drivers and see if my video card running okay?  I had 10.04 installed on my Thinkpad r61 with Compiz Manager and life was great, until the machine took water damage. I bought an ASUS (X54H) since and am trying out 12.04, but the desktop just doesn't look right.  I always struggled with video driver installation. There are no proprietary drivers available in the hardware manager.  When I run lspci | grep VGA:  00:02.0 VGA compatible controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   I will greatly appreciate your help. I want to use Linux more, but like I said video drivers appear to be my biggest concern. I have also tried 12.04 on my desktop PC, but again failed to configure video card, so switched back to Windows 7. Does $\pi$ contain all possible number combinations? What are the limitations of healing magic? While reading Sorcerer's Stone, I just finished the chapter where Harry walks in on Snape and Filch bandaging Snape's leg.  In a previous chapter Neville falls off of his broom and we see that magic can easily heal his broken wrist.  If magic can heal an injury like broken bones, why wouldn't Snape heal his leg, which was just torn up by a 3-headed dog?  My initial thought was that only certain people specialize in healing magic, but while poking around this site I came across this question:   This shows that Snape had at least some healing abilities.  So why didn't he heal his own leg?  Are there limitations that prevented Snape from healing his own wounds? Nested cross validation for model selection How can one use nested cross validation for model selection?   From what I read online, nested CV works as follows:   There is the inner CV loop, where we may conduct a grid search (e.g. running K-fold for every available model, e.g. combination of hyperparameters/features) There is the outer CV loop, where we measure the performance of the model that won in the inner fold, on a separate external fold.    At the end of this process we end up with $K$ models ($K$ being the number of folds in the outer loop). These models are the ones that won in the grid search within the inner CV, and they are likely different (e.g. SVMs with different kernels, trained with possibly different features, depending on the grid search).  How do I choose a model from this output? It looks to me that selecting the best model out of those $K$ winning models would not be a fair comparison since each model was trained and tested on different parts of the dataset.  So how can I use nested  CV for model selection?   Also I have read threads discussing how nested model selection is useful for analyzing the learning procedure. What types of analysis /checks can I do with the scores that I get from the outer K folds? Whether the players killed a mage and got his spellbook(s) or they met with a nice wizard who let him or her learn a spell from his spellbook, can a player prepare a spell from a spellbook that isn't his or hers?  I'm asking because the PHB p. 114 says:     [...] Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.    emphasis mine   Further, the block on the left as a Copying a Spell into the Book ("the" Book as in "your" Book, since it appears in the block titled Your Spellbook), which explains how you can copy a spell from one book to another, assuming the book is from a stranger, it cost more and takes more time. It actually says:     [...] deciphering the unique system of notation used by the system of notation of the wizard who wrote it. [...]   It continues saying that you have to learn the various parts of the spell (verbose, material, gesture.)  This makes it sounds like a wizard would have a really hard time to prepare a spell from another's spellbook. Is that the case?