Let $A$, $B$ be closed subsets of topological space $X$. Given that $A\cup B$ and, $A\cap B$ are both connected.  How can we show that both A, B are connected? This is in general not true for any subsets of $X$. I can't construct a counter example.  I mean,  both $A$, $B$ must be closed for this. What happens when they are open?
Suppose that $V, W$ are closed in $X$ and that $V\cap W, V\cup W$ are connected. Show that $V$ and $W$ are connected.  My attempt: Suppose that $V=U_1\cup U_2$ where $U_i$ are open in $V$, not empty and $U_1\cap U_2=\emptyset$. This induces a separation for $V\cap W$: $$ V\cap W = (U_1\cap W)\cup(U_2\cap W).$$  $U_i\cap W$ is open in $V\cap W$  $(U_1\cap W)\cup (U_2\cap W)=\emptyset$ as otherwise $U_1\cap U_2 \ne \emptyset$.  Now, I need to show that both sets are not empty. We know that $V\cup W$ is connected and closed in $X$. So, for all $x\in V\cup W=U_1\cup U_2\cup W$ and all neighborhoods $Z$ (in $X$!) of $x$ we have that $Z\cap (V\cup W) \ne \emptyset$. We can choose $x\in U_1 \subseteq V\cup W$. But then I would have to find a neighborhood of $x$ in $X$. I can't find one... How can I complete this bullet point?   Thanks.
With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons? centos: using epel repository on a server machine Are there any downsides in enabling epel repository on a server machine or is it something every sysadmin does? Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$.  What I would like to show is the converse, that is,      if $\text{tr}(A^n)=0$ for all positive integers $n$, then $A$ is nilpotent.   I tried to show that $0$ must be an eigenvalue of $A$, then try to show that all other eigenvalues must be equal to 0. However, I am stuck at the point where I need to show that $\det(A)=0$.  May I know of the approach to show that $A$ is nilpotent? Inertial frames of reference I'm struggling with the notion of an inertial frame of reference. I suspect my difficulty lies with the difference between Newtonian and relativistic inertial frames, but I can't see it.   I've read that Newton's laws apply in any non accelerating frame of reference, which are called inertial frames. So, if I play pool on a train moving with uniform velocity, the balls behave in the same way as if I were playing pool in a pool hall. So the train is (to a good approximation, ignoring forces caused by the Earth's rotation and movement around the Sun etc) an inertial frame.   I've also read that in an inertial frame, Newton's first law is satisfied. So, if I slide a rock on a sheet of ice and I could somehow eliminate the frictional forces between the rock and the ice, the rock would carry on sliding for ever, as predicted by Newton's first law.   Question 1 - is the  sheet of ice also therefore a good approximation to an inertial frame?  Question 2 - are these two definitions of inertial frames saying the same thing in different ways?  Both these examples occur in gravitation fields, which doesn't seem to matter as both the train and the ice sheet are presumably good approximations to inertial frames.  Question 3 - is gravity irrelevant when defining these two inertial frames?  In special relativity, I've read (Foster and Nightingale) that an inertial frame is also one where Newton's first law holds. But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity.  Question 4 - how come you can use the same definition of an inertial frame (satisfied Newton's first law) but in the case of special relativity, the train and the ice sheet aren't inertial frames. Is this to do with not being able to synchronize clocks in a gravitational field?  In general relativity, I've read (Schutz) that a freely falling frame is (on Earth) the only possible (and local) inertial frame. So again the first two examples would not be inertial frames in general relativity.  Question 5 - am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?  Question 6 - have I missed anything else that might be useful?  Many thanks  Edit. On reflection, am I right in thinking that because Newtonian mechanics assumes universal, absolute time we don't need to worry about synchronizing clocks in a Newtonian inertial frame. Therefore we don't need to worry about gravity in a Newtonian inertial frame, because in such a frame gravity does not affect time.  This is not the case in spacetime, as here gravity does affect time and the only way to have synchronized clocks in an inertial frame in spacetime is: 1. have no gravitational field, or 2. use a local, freely falling frame.  Am I on the right track here? Meta Meta Privilege I just took a look at meta today and noticed something weird in the meta privileges. If you look at the meta privilege list, you can see that at . I know that there is no meta for meta, and the link they give you in the privilege page leads you to meta.stackexchange.com, right where you started. Why do they have this privilege? I have a feeling that this is a bad recursive joke. You must be  member of meta to get the privilege of being a member of meta! :) Use physical harddisk in Virtual Box Can an abstract class have a constructor?  If so, how can it be used and for what purposes? Under the U.S.'s Copyright Law, is it an act of copyright infringement if we changed the color of another artist's image to black and white? If we have a point charge and outside of it we have a non-conducting Gaussian sphere, then Gauss's law says that the net flux should be zero. I agree that the total field lines coming in are equal to the number of field lines coming out, but isn't it true that the magnitude of the electric field at those segments of the sphere where the field is coming in is greater than those where it is coming out? So when we integrate, wouldn't these areas, having a greater electric field, contribute more to the net flux? I don't see why they would cancel each other out. Showing that two topologies give rise to the same topology Can one construct a non-measurable set without Axiom of choice? Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice?  Related question on MO says it is consistent: Interpretation of ROC curve I have a ROC curve like the figure below:   What can be inferred given this kind of curve in general?  What are the differences of states 1, 2 and 3 in the figure?  What causes the diagram to jump straight from state 1 to state 3? Most of the ROC curves on the internet look like the figure below: Using "you" and "your" as a representation for yourself and everyone in general Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. Let's say I have the domain johnsmith.com and I want people to reach me via email, what is the best email username to use?  Some examples:   john@johnsmith.com - redundant john.smith@johnsmith.com - redundant and looks kinda stupid mail@johnsmith.com - makes sense, but looks very "technical" me@johnsmith.com root@johnsmith.com - for the geeks among us ;-)   Of course, technically I can simply say that I will be able to receive *@johnsmith.com, however, for a better user experience I still just want one email address for public presence (website, business cards, etc.).  Any suggestions? Which form should be used for at­tribu­tive nouns like “stu­dent union”: sin­gu­lar or plu­ral, or pos­ses­sive sin­gu­lar or pos­ses­sive plu­ral? When should a noun that’s used at­tribu­tively to de­scribe an­other  noun be plu­ral, and when should it be sin­gu­lar?  And when should it be pos­ses­sive, like bak­er’s dozen and when should it be plu­ral pos­ses­sive, like farm­ers’ mar­ket?  In other words, why do we say teach­ers union rather than teacher  union?  And why do we say wed­ding plan­ner rather than wed­dings plan­ner?  Which of these vari­ants is or are cor­rect?        stu­dent union   stu­dents union   stu­den­t’s union   stu­dents’ union      What about com­mu­nity val­ues ver­sus com­mu­ni­ty’s val­ues?  Please note I am look­ing for a gen­eral rule or at least some tips.  These are only ex­am­ples.  Up­date: It seems even na­tive speak­ers fol­low their per­sonal  style to write such com­pound words. I won­der why in the IELTS lis­ten­ing sec­tion, the lan­guage learner has to write a spe­ci­fic form and there is no rule for this. The following example works as expected:  \documentclass{article} \usepackage[table]{xcolor} \usepackage{makecell, multirow}     \renewcommand\theadfont{\normalsize\bfseries\color{white}}     \renewcommand{\theadgape}{} \usepackage{hhline}  \begin{document} \centering     \setlength\arrayrulewidth{2pt}     \arrayrulecolor{yellow}     \def\clinecolor{\hhline{&gt;{\arrayrulecolor{yellow}}--%                             &gt;{\arrayrulecolor{gray}}-}}     \setlength\extrarowheight{2pt} \begin{tabular}{ c | c | c }%     \rowcolor{gray} \multicolumn{2}{c|}{\thead{AAA -- BBBB}}    &amp;                                   \\     \clinecolor     \rowcolor{gray} \color{white} AAA   &amp;   \color{white} BBB   &amp;   \multirow{-2}{*}{\thead{CCC}}   \\     \hhline{&gt;{\arrayrulecolor{yellow}}-|-|-}     \rowcolor{gray!25} aaa                 &amp;   bbb                 &amp;   ccc                             \\     \hline     \rowcolor{gray!25} ddd             &amp;   eee                 &amp;   fff     \end{tabular} \end{document}     But the following not:  \documentclass[xcolor=table]{beamer} \usepackage{makecell, multirow}     \renewcommand\theadfont{\normalsize\bfseries\color{white}}     \renewcommand{\theadgape}{} \usepackage{hhline}  \begin{document} \begin{frame} \frametitle{test} \centering     \setlength\arrayrulewidth{2pt}     \arrayrulecolor{yellow}     \def\clinecolor{\hhline{&gt;{\arrayrulecolor{yellow}}--%                             &gt;{\arrayrulecolor{gray}}-}}     \setlength\extrarowheight{2pt} \begin{tabular}{ c | c | c }%     \rowcolor{gray} \multicolumn{2}{c|}{\thead{AAA -- BBBB}}    &amp;                                   \\     \clinecolor     \rowcolor{gray} \color{white} AAA   &amp;   \color{white} BBB   &amp;   \multirow{-2}{*}{\thead{CCC}}   \\     \hhline{&gt;{\arrayrulecolor{yellow}}-|-|-}     \rowcolor{gray!25} aaa                 &amp;   bbb                 &amp;   ccc                             \\     \hline     \rowcolor{gray!25} ddd             &amp;   eee                 &amp;   fff                             \\     \hhline{|-|-|-|}     \end{tabular} \end{frame} \end{document}     Why are the in beamer rules so strangely misplaced? Is there some workaround? When running Internet Explorer from the the modern UI, I can't find out how to access tabs.  I can create tabs using Ctrl + T, but I'm not sure how to switch between tabs, see what tabs I have open, etc.  How do I navigate between tabs when using the modern UI Internet Explorer? Top Down RPG Movement w/ Correction? I would hope that we have all played Zelda: A Link to the Past, please correct me if I'm wrong, but I want to emulate that kind of 2D, top-down character movement with a touch of correction. It has been done in other games, but I feel this reference would be the easiest to relate to.  More specifically the kind of movement and correction I'm talking about is:   Floating movement not restricted to tile based movement like Pokemon and other games where one tap of the movement pad moves you one square in that cardinal direction. This floating movement should be able to achieve diagonal motion. If you're walking West and you come to a wall that is diagonal in a North East/South West fashion, you are corrected into a South West movement even if you continue holding left (West) on the controller. This should work for both diagonals correcting in both directions. If you're a few pixels off from walking squarely into a door or hallway, you are corrected into walking through the hall or down the hallway, i.e. bumping into the corner causes you to be pushed into the hall/door.   I've hunted for efficient ways to achieve this and have had no luck. To be clear I'm talking about the human character's movement, not an NPC's movement. Are their resources available on this kind of movement? Equations or algorithms explained on a wiki or something?  I'm using the XNA Framework, is there anything in it to help with this? Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Parse JSON using Python?