Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?
Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf   implies that $X$ is separable.   My attempt:  Let $d$ be the metric that induces the topology $\mathcal{T}$.   For every $n \geq 1$, consider the open cover: $$\{B_X(x,1/n)\mid x \in X\}$$  Because the space is Lindelöf, this implies that for every $n \geq 1$, there is a countable (or finite) subcover given by $$\mathcal{B}_n:= \{B_X(x,1/n) \mid x \in I_n, |I_n| \leq |\mathbb{N}|\}$$  Now, define $D:= \bigcup_{n\geq 1} I_n$. Let $y \in X$. Let $\epsilon &gt; 0$. Choose $m$ a positive integer so large that $1/m &lt; \epsilon$.  Because $y\in X =\bigcup\mathcal{B}_m$, it follows that there exists $x \in D$ such that $y \in B_X(x,1/m)$, meaning that $d(x,y) &lt; 1/m &lt; \epsilon$, such that $x \in B_X(y, \epsilon) \cap D$. Hence, $y \in \overline{D}$ and it follows that $D$ is a dense countable set in $X$, such that $X$ is separable.  Is this correct?
Find the all possible real solutions of $x^y=y^x$ Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers.  My attempt:I observed the following solutions  $x=2,y=4$  $x=4,y=2$  $y=x$  Is there any other possible solutions? First of all, I think this question is not duplicated, because I have tried all other solutions I found here, but nothing solved my problem.  I use Ubuntu 14.04 64 bit.  I have tried until now: sudo apt-get install skype -> It didn't work. The error was:   The following packages have unmet dependencies: skype : Depends: skype-bin E: Unable to correct problems, you have held broken packages.   I have tried sudo dpkg add-architecture &amp;&amp; sudo dpkg -i skype-ubuntu-precise_4.3.0.37-1_i386.deb but it didn't work either. The error was:  dpkg: dependency problems prevent configuration of skype: skype depends on libqt4-dbus (&gt;= 4:4.5.3); however: Package libqt4-dbus is not installed. skype depends on libqt4-network (&gt;= 4:4.8.0); however: Package libqt4-network:i386 is not installed. skype depends on libqt4-xml (&gt;= 4:4.5.3); however: Package libqt4-xml:i386 is not installed. skype depends on libqtcore4 (&gt;= 4:4.7.0~beta1); however: Package libqtcore4:i386 is not installed. skype depends on libqtgui4 (&gt;= 4:4.8.0); however: Package libqtgui4:i386 is not installed. skype depends on libqtwebkit4 (&gt;= 2.2~2011week36); however: skype depends on libstdc++6 (&gt;= 4.2.1); however: Package libstdc++6:i386 is not installed. skype depends on libx11-6; however: Package libx11-6:i386 is not installed. skype depends on libxext6; however: Package libxext6:i386 is not installed. skype depends on libxss1; however: Package libxss1:i386 is not installed. skype depends on libxv1; however: Package libxv1:i386 is not installed. skype depends on dpkg: error processing package skype (--install): dependency problems - leaving unconfigured Processing triggers for hicolor-icon-theme (0.13-1) ... Processing triggers for gnome-menus (3.10.1-0ubuntu2) ... Processing triggers for desktop-file-utils (0.22-1ubuntu1) ... Processing triggers for bamfdaemon (0.5.1+14.04.20140409-0ubuntu1) ... Rebuilding /usr/share/applications/bamf-2.index... Processing triggers for mime-support (3.54ubuntu1) ... Errors were encountered while processing: skype   I have tried to use after sudo apt-get -f install but it didn't solve the issue.  I have tried to install by using Gdebi. The error was:  Cannot install 'libqtwebkit4:i386'   I have tried to install using Ubuntu Software Center. The error was:  Cannot install 'libqtwebkit4:i386'   I have tried changing the server in "Software and Updates". I have tried to use "Main Server" and "Servers for United States". But it couldn't solve my problem.  Anyone has any idea how to solve that?  SOLVED  Well, I formatted my pc and I installed Skype first of all. Probably another installation that I had done before the Skype installation was the problem, i guess.  Thanks people! Hotspot and Internet access together — why not? If I try to set up a hotspot in GNOME, I get the following message:  If you have a connection to the Internet other than wireless, you can use it to share your internet connection with others. Switching on the wireless hotspot will disconnect you from &lt;network name&gt;. It is not possible to access the internet through your wireless while the hotspot is active.  What are the technical reasons why I can't set up an Hotspot and be connected to the Internet at the same time? Why don’t photons interact with the Higgs field and hence remain massless? Vote for Tag Synonyms Directly on the Page Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. I was about to add an extra signal handler to an app we have here and I noticed that the author had used sigaction() to set up the other signal handlers. I was going to use signal(). To follow convention I should use sigaction() but if I was writing from scratch, which should I choose? How can I ensure my exit from the US by land is recorded? Seeing as the US does not perform border control on exit (save for occasional spot checks, usually when traffic is moving slowly at the border), how do I make sure my exit by land is recorded so that I'm not incorrectly flagged as an overstayer?  When exiting the US by air or sea, this is not a problem, as airlines and shipping companies send passenger records to US authorities. What about exiting by land though? What's the difference between a projection and a datum? Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold. What does the dot mean relation between $R^2$ of simple regression and multiple regression A very basic question concerning the $R^2$ of OLS regressions   run OLS regression y ~ x1, we have an $R^2$, say 0.3 run OLS regression y ~ x2, we have another $R^2$, say 0.4 now we run a regression y ~ x1 + x2, what value can this regression's R squared be?   I think it's clear the $R^2$ for the multiple regression should be no less than 0.4, but is it possible for it to be more than 0.7? Suppose $K \subset L$ is a finite field extension of degree $m$. Is it true that there exists some natural $n$ such that $L$ is isomorphic to a subfield of $M^{n\times n}(K)$, the ring of $n\times n$ matrices with entries in $K$? What is the smallest dimension (as a function of $m$) for which an isomorphism exists?  The example I had in mind was the 2-dimensional extension $\mathbb{R} \subset \mathbb{C}$ with $\mathbb{C}$ being isomorphic to $\{ \begin{pmatrix}a&amp;-b\\b&amp;a\end{pmatrix} : a, b \in \mathbb{R}\}$. How to properly typeset multiletter quantities, e.g. Re or Nu (Reynolds, Nusselt numbers) Is there a canonical way to typeset such quantities in equations and inline math?  \documentclass{article} \begin{document} $Re$, $Nu$ are clearly not an option. $\mathit{Re}$, $\mathit{Nu}$ appear ok to me. \end{document} The formula for a one sample t-test, as given by  is  $t = \dfrac{\bar{x} - \mu_0}{s\big/\sqrt{n}}$  However, in this equation, if the sample standard deviation, $s$, is zero, $t$ will be undefined. Is there any reasonable way to get a $t$ value in this case? Clerical mistake in UK visa application: should I make a pen-and-ink correction or electronically edit the PDF? I've gone through all similar questions but I believe my specific question doesn't have an answer yet.  I've applied for a Standard Visitor visa. After I completed the application, made an appointment and paid the fee, I noticed that the way I spelled my name on the application differs from how it is spelled in the passport by 1 letter (this is partially because of myself being an idiot, but mainly because my new passport has an incorrect transliteration of my name, whereas my old passport and any other document in my life has the correct one).  So here are the options I see:   Print the application and make a pen-and-ink correction with a black ballpoint pen by crossing the printed name/letter and writing the correct one - as answers to other similar questions suggests. Edit the PDF electronically and print it so that there are no visible corrections. However, as the form is saved at their end, there will be a discrepance between the printed form and saved form with no visible signs of correction.   Which way has better chance for me to have my application accepted?  Also (probably doesn't matter much but still): if it's the pen-and-ink correction, is it better to cross-out the whole name or only the wrong letter? How should I apply brick to an interior wall? ...where each object also has references to other objects within the same array?  When I first came up with this problem I just though of something like   var clonedNodesArray = nodesArray.clone()   would exist and searched for info on how to clone objects in javascript. I did find a  on StackOverflow (answered by the very same @JohnResig) and he pointed out that with jQuery you could do  var clonedNodesArray = jQuery.extend({}, nodesArray);   to clone an object. I tried this though, this only copies the references of the objects in the array. So if I   nodesArray[0].value = "red" clonedNodesArray[0].value = "green"   the value of both nodesArray[0] and clonedNodesArray[0] will turn out to be "green". Then I tried  var clonedNodesArray = jQuery.extend(true, {}, nodesArray);   which deep copies an Object, but I got "too much recursion" and "control stack overflow" messages from both Firebug and Opera Dragonfly respectively.  How would you do it? Is this something that shouldn't even be done? Is there a reusable way of doing this in Javascript? How to remove the handles from this faucet My bathroom faucet is leaking around the base of my right handle.  With previous faucets I've been able to use a set screw to loosen the handle and replace the washer.  With these, though, I can't seem to pop off the faucet handle.  Does anyone know how to remove this handle?  Does anyone know the brand of this faucet?  Update -- Nov 29th  I've already tried to get those cap ends off.  I tried to use some pliers and unscrew them, but couldn't remove it.  I tried taking a small flat head and popping off the ends.  Still nothing.  It doesn't seem like the come off.  I also tried unscrewing the handles, but they just don't see to turn for me.  Also there is definitely no set screw on the handles.  I've searched all around the faucet and found nothing indicating a set screw. Distribution of orbital velocities in a disk galaxy for N-body simulation? I'd like to write an N-body simulation in which I collide two disk galaxies. To give you an idea of the accuracy I'm trying to achieve, I'm aiming to make this my screensaver at 30fps on my work desktop. Think Euler's method, large epsilons in denominators to prevent blow up, all that lazy stuff.   One place I don't want to be lazy, however, is in the initial construction of the galaxies I'm trying to collide. I want to design a galaxy that will at least sort of hold together by its own gravity until it smash it into another one.   How might I go about selecting the orbital velocities for each of the bodies composing the galaxy? How to restart X Window Server from command line? How can I restart X Window Server from the command line?  I'd really like to be able to restart my GUI without having to do a full system reboot.