Let $X$ be a connected metric space, let $A \subseteq X$ be a connected set. Let $\mathcal{C}$ be a connected component of $X-A$. Show that $X-\mathcal{C}$ is connected.   Ok, so Im been dealing with this exercise for a while, and I am completely stucked.   I've managed the following (which I don't know if it is even true.)  Since $X-A \subseteq \mathcal{C}$ we have that $X-\mathcal{C} \subseteq A$. Now since $A$ is a connected set, and $\mathcal{C}$ is closed (since it's a connected component), one way to prove the statement is to see that $X-\mathcal{C}$ is closed in $A$ because in this case, $X-\mathcal{C}$ would be open and closed in $A$, connected which implies $A=X-\mathcal{C}$ and therefore $X-\mathcal{C}$ is connected.   However Im not being able to show that $X-\mathcal{C}$ is closed in $A$, (im not sure if in fact it is).   If not, If its possible, I would like a hint, not a complete solution!
I have an exercise found on a list but I didn't know how to proceed. Please, any tips?  Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of $M\setminus X$ that $M\setminus C$ is connected.
I hear a lot of talk about apparmor, I want to know the following:    What is apparmor? How does apparmor work? Fold a plane. then fold again Imagine a piece of paper and you fold it in half and then fold it again. that's what I am trying to do and cant. If I use bones the influence of the bones won't allow me to because I need 2 bones to control the same area. If I try shape keys it just deforms the mesh if I try to make it in multiple steps or does the wrong animation because it draws the shortest path instead of rotating from a point like a piece of paper would. Can blender use the Alpha channel of an image as transparency on a plane? R linear regression categorical variable "hidden" value In a simple battery + resistor circuit, what form of energy is lost from the electrons upon exiting the resistor? I will give this question a little context. Firstly, as I understand it, as soon as I "close the switch" on a circuit, electric current pretty quickly establishes a steady state where, at any given cross section along a wire, the average kinetic energy of that slice is both constant and equal anywhere along the wire. (Please correct me if I am misunderstanding).   If the average kinetic energy is constant anywhere throughout the wire in the circuit (which confuses me, as I would think that the electrons should all accelerate after exiting the resistor since they are no longer being impeded as much), that means that the average velocity is constant. So, in the example of a simple circuit with a battery of v Volts and a resistor of o Ohms, my question is the following:  Because a 'voltage drop' is known to occur across the resistor, what type energy is being "traded" to generate the heat that radiates from the resistor? I would think that the reflexive answer is "electrical energy"...hence the VOLTAGE drop (the sacrificed energy is clearly not kinetic, as the velocity is constant everywhere). However, I find this confusing. Does this mean that if I had a positive test charge, it would be easier for me to bring it to the beginning of the resistor as compared to bringing it to the end of the resister?     Further, if the correct answer IS "electrical energy", why exactly DOES electrical energy get "dissipated" as the electrons pass through the resistor? I always here the comment Oh! It's because all the electrons are running into densely packed crap which impedes their flow but that to me sounds like a reason for their KINETIC energy to be reduced. However, clearly that's not the case. So, ultimately, I guess the real question is:  **What goes on in the resistor that is literally removing electrical energy **. I am sure that this is a quantum mechanics question, but my quantum mechanics knowledge is not terribly strong. If I could get an answer that is devoid of crazy wave function equations, I would greatly appreciate it. Thanks! I sometimes get files from my clients that have the wrong file extension. For example, the name is image.jpg but the file is actually a TIFF image. In many cases I can clarify it by opening the file in a text editor, looking at the first few bytes, then deducing which file type it is.  This works for me with JPEG, TIFF, GIF and PDF files. However there are many more file types out there.  Is it possible to automate identification of the correct file type by analyzing the containing data? polar decomposition of a square A when A is not full rank/invertible I'm currently studying  and Python programming. I just want to know how I would go about executing a Linux command in Python. The commands I want to execute are:  echo 1 &gt; /proc/sys/net/ipv4/ip_forward iptables -t nat -A PREROUTING -p tcp --destination-port 80 -j REDIRECT --to-port 8080   If I just use print in Python and run it in the terminal will it do the same as executing it as if you was typing it yourself and pressing Enter? He has  yet to receive an appointment.   He is yet to receive an appointment.      Is there any difference in meaning? Is one more correct than the other? I go to the Minecraft menu and click on singleplayer to test out the 1.9 update. I made a new world (no mods), it said downloading terrain and then it crashes every time (I've tried about 20 times now).  Has anyone else experienced similar issues and know what the fix may be?  Here is the launcher log:  Completely ignored arguments: [--nativeLauncherVersion, 301] [00:35:12] [Client thread/INFO]: Setting user: Teddy_Cromwell [00:35:12] [Client thread/INFO]: (Session ID is &lt;censored&gt;) [00:35:14] [Client thread/INFO]: LWJGL Version: 2.9.4 [00:35:14] [Client thread/WARN]: Removed selected resource pack resources (no region and battle music).zip because it's no longer compatible [00:35:14] [Client thread/INFO]: Reloading ResourceManager: Default [00:35:15] [Sound Library Loader/INFO]: Starting up SoundSystem... [00:35:16] [Thread-5/INFO]: Initializing LWJGL OpenAL [00:35:16] [Thread-5/INFO]: (The LWJGL binding of OpenAL.  For more information, see http://www.lwjgl.org) [00:35:16] [Thread-5/INFO]: OpenAL initialized. [00:35:16] [Sound Library Loader/INFO]: Sound engine started [00:35:18] [Client thread/INFO]: Created: 1024x512 textures-atlas [00:35:26] [Server thread/INFO]: Starting integrated minecraft server version 1.9 [00:35:26] [Server thread/INFO]: Generating keypair [00:35:27] [Server thread/INFO]: Preparing start region for level 0 [00:35:28] [Server thread/INFO]: Preparing spawn area: 5% [00:35:29] [Server thread/INFO]: Preparing spawn area: 8% [00:35:30] [Server thread/INFO]: Preparing spawn area: 15% [00:35:31] [Server thread/INFO]: Preparing spawn area: 21% [00:35:32] [Server thread/INFO]: Preparing spawn area: 27% [00:35:33] [Server thread/INFO]: Preparing spawn area: 34% [00:35:34] [Server thread/INFO]: Preparing spawn area: 42% [00:35:35] [Server thread/INFO]: Preparing spawn area: 48% [00:35:36] [Server thread/INFO]: Preparing spawn area: 56% [00:35:37] [Server thread/INFO]: Preparing spawn area: 66% [00:35:38] [Server thread/INFO]: Preparing spawn area: 74% [00:35:39] [Server thread/INFO]: Preparing spawn area: 81% [00:35:40] [Server thread/INFO]: Preparing spawn area: 91% [00:35:41] [Server thread/INFO]: Preparing spawn area: 98% [00:35:42] [Server thread/INFO]: Teddy_Cromwell[local:E:6d7525a7] logged in with entity id 301 at (244.5, 69.0, 244.5) [00:35:42] [Server thread/INFO]: Teddy_Cromwell joined the game # # A fatal error has been detected by the Java Runtime Environment: # #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=0x00007fffcf6e6b37, pid=15456, tid=12424 # # JRE version: Java(TM) SE Runtime Environment (8.0_25-b18) (build 1.8.0_25-b18) # Java VM: Java HotSpot(TM) 64-Bit Server VM (25.25-b02 mixed mode windows-amd64 compressed oops) # Problematic frame: # C  [ig8icd64.dll+0x16b37] # # Failed to write core dump. Minidumps are not enabled by default on client versions of Windows # # An error report file with more information is saved as: # C:\Users\Connor\AppData\Roaming\.minecraft\hs_err_pid15456.log # # If you would like to submit a bug report, please visit: #   http://bugreport.sun.com/bugreport/crash.jsp # The crash happened outside the Java Virtual Machine in native code. # See problematic frame for where to report the bug. # AL lib: (EE) alc_cleanup: 1 device not closed Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release Tags on badge question list I'm trying to get the Weed-Eater hat on Stack Overflow. To do so, I'm looking at the list of questions . The problem is, no tags show up there, and so it's rather difficult to see which ones I have a shot at. Any chance of showing the tags on this page?  I realize that getting this before Winter Bash is over would be difficult, but it would be good to keep such things in mind for the future. Where does matter go after reaching singularity in uncharged black holes? I am a layman in physics and just read about black holes on the internet. I read that matter encounters geodesic incompleteness in the singularity in an uncharged black hole.  I heard an analogy of geodesic incompleteness as a straight line on a paper reach a hole on the paper, so it cannot continue. But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)? So, if matter reaches the singularity, is it possible too (to reach another dimension)?   I also heard that the matter is annihilated when reaching the singularity, does it mean it disappear from this world, and violates conservation of energy? Password protecting a page I'm trying to password protect a page but it doesn't seem to be working. I've set the password for the page but when you navigate to it it just loads up like normal and doesn't ask for any passwords.  This is the loop I have on my page:  &lt;?php /**  * @package WordPress  * @subpackage Default_Theme  * Template Name: Page  */  get_header(); ?&gt;  &lt;?php global $post; if ( post_password_required( $post ) ) { ?&gt;  &lt;?php $fields = get_acf(); ?&gt;  &lt;script type="text/javascript"&gt;  (function() {   window.onload = function() {   ...  }} )  &lt;/script&gt;  &lt;div id="map"&gt;&lt;/div&gt;  &lt;?php } else { ?&gt;     Not passworded        &lt;?php if (have_posts()) : ?&gt;         &lt;?php while (have_posts()) : the_post(); ?&gt;          &lt;?php endwhile; ?&gt;     &lt;?php endif; ?&gt;         &lt;?      } // Page isn't password-protected     ?&gt; &lt;?php get_footer(); ?&gt;   I want to password protect everything within the page.  UPDATE:  I have custom content on the page so if possible I need to be able to wrap some code around it all.  Update: I am logging in to my Ubuntu Server using my username. Once I am logged in I am typing passwd command. Entering a new password but a second after getting following error messages:  passwd: Authentication token manipulation error passwd: password unchanged   What is wrong here? How can I change my password otherwise if I don't have access to that server physically, i.e. I am connecting remotely with ssh using terminal. Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers. Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers.   This is question 5 from Herstein Page 35. I have proved that conclusion holds for three positive integers.Thanks. Retail boxed Steam activated games: can they be resold? I recently purchased retail boxed versions of   Dead Island (which is awesome) Warhammer 40k: Space Marine (not so much)   I finished both games and I was thinking of reselling them or giving them to a friend to play. However, these are both Steam activated games that had a required Steam registration code I had to enter -- and validate on Steam -- as part of the installation.  If I give these games to a friend or resell them, will they even work? Is the registration code single use and tied to a single person / computer / Steam account? Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$? Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that $\lim\limits_{n \to\infty}a_{n+1} = \lim\limits_{n\to\infty}a_n$? If the matrix $A^3 = 0$, is $A$ singular? I've been Googling around with no luck.  I'm looking for a way to do getc() or gets() or whatever that does not echo to the terminal.  I saw kbhit() but this doesn't appear to be part of ANSI.  Ideally, I'd like code that looks like  char s[100]; no_echo_gets(s); /* Won't echo to screen while being typed */ printf("%s\n", s);   Does anybody know a good ANSI compliant way to do this? A word for "mercy killing" "Mercy killing" is an act of killing someone who is already dying and in terrible pain, in order to reduce his or her suffering.    Is there any single word for "mercy killing"?