Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.    (Rudin's Principles of Mathematical Analysis, 3rd ed., Chapter 2, Problem 28)  Now I've managed to hit upon a proof using the preceding problem (which can be generalised) as follows:   Let $X$ be a separable metric space, $E \subset X$, $E$ is uncountable, and let $P$ be the set of all condensation points of $E$. Prove that $P$ is perfect and that at most countably many points of $E$ are not in $P$. In other words, show that $P^c \cap E$ is at most countable.   Now what bothers me is what Rudin states as the corollary to the result he asks us to prove in Problem 28. Here's what he states:   Every countable closed set in $\mathbb{R}^k$ has isolated points.   How to prove this from the original result?   Can we also state the following?   Every countable closed set in a separable metric space has isolated points.
I'm reading Baby Rudin and exercise 28 of chapter 2 reads "Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable." I don't understand why the set must be closed in order for this to be true.  My proof is as follows, and I don't make any reference to the closedness of the set:  Let $E$ be a set in a separable metric space.  Let $B$ be a countable base of that metric space.  Let $P$ be the set of condensation points of $E$.  I claim that $P$ is perfect and $E-P$ is at most countable.  If $p$ is a limit point of $P$, then every neighborhood of $p$ includes at least one point $q$ of P.  For a neighborhood $N_{p}$ of $p$, choose a neighborhood $N_{q} \subset N_{p}$ of $q$.  Since $q$ is a condensation point of $E$, uncountably many points of $E$ are in $N_{q}$, and thus in $N_{p}$.  So $p\in P$ and $P$ is closed.  If $p \in P$, then $p$ is a condensation point of $E$, so every neighborhood $N_{p}$ of $p$ must include uncountably many points of $E$.  If $p$ is not a limit point of $P$, then there exists a neighborhood $N_{p}$ of $p$ such that $N_p$ includes no condensation points of $E$ except for $p$.  This means that every point $q\in N_{p}-\{p\}$ must have a neighborhood $N_{q}$ that does not have uncountably many points of $E$.  For each $b_{k}\in B \cap \mathscr{P}(N_{p})$, choose such a neighborhood $N_{q_{k}} \supset b_{k}$ of some point $q\in N_{p}-\{p\}$.  Now $\{N_{q_{k}}\}$ covers $N_{p}$, and $\bigcup_{k=1}^{\infty}N_{q_{k}}\supset N_{p}$ has at most countably many points of $E$, for it is a union of countably many sets that have at most countably many points of $E$.  This contradicts the assumption that $p\in P$.  So $P$ is perfect.  Let $S=E-P$. There are no condensation points in $S$.  So there exists an open cover $\{N_s\}$, where each $N_s$ is a neighborhood of $s \in S$ with at most countably many elements of $S$. For each $b_{k}\in B \cap \mathscr{P}(S)$, choose such a neighborhood $N_{s_{k}} \supset b_{k}$ of some point $s\in S$.  Then we have a countable subcover $\{N_{p_{k}}\}$ of $S$. As $N_{p_{k}} \cap S$ is at most countable for all $k$, $S$ must be at most countable.  So what am I doing wrong?  Or is Rudin's assumption of closedness extraneous?
Simple past or present perfect? Which is preferable, simple past or present perfect, in situations like this one: I had a conversation with my friend and we planned to do something; shortly afterwards I call him back to announce the accomplishments. My biggest doubt about simple past is that perfect aspect is used to indicate past action (accomplishment) with current relevance, at the same time I'm not sure if I'm allowed to use present perfect, because there's a gap between accomplishing and the current moment, which is a characteristic of simple past. Why shouldn't I prefix my fields? I've never been a fan of Hungarian notation, I've always found it pretty useless unless you're doing some really low level programming, but in every C++ project I've worked on some kind of Hungarian notation policy was enforced, and with it the use of some 'not-really-Hungarian' prefixes as m_ for fields, s_ for statics, g_ for globals and so on.  Soon I realized how much useless it was in C# and gradually started to drop all of my old habits... but the 'm_' thing. I still use the m_ prefix on private fields because I really find it very useful to being able to distinguish between parameters, locals and fields.  The  says I shouldn't, but it does not say why (the way e.g. Google's conventions generally tend to rationalize their prescriptions).  Are there reasons why I shouldn't or is it only a matter of style. If it is the latter, are prefixes generally considered a bad style and can I expect negative reactions from other people working on the codebase? iPhone, reproduce the magnifier effect Why is my document pdf generated by pdflatex looks different from the one generated by latex+dvips+ps2pdf ? From the code below, i have tow different papers switch to the tow cited commands above :  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[francais]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \usepackage{lipsum} \author{M ROUAN SERIK} \title{Article with \LaTeXe} \fontfamily{cmr} \begin{document} \maketitle \section{Titled section} \lipsum \end{document}   Here's my pdffonts for the tow pdfs : Pdflatex:    name                   type     encoding  emb sub uni object ID   ------------------------------- ---------------- --- --- --- ---------   AMTRHA+SFRM1728        Type 1    Custom   yes yes no       4  0   DPTCLU+SFRM1200        Type 1    Custom   yes yes no       5  0   JVZRKA+CMMI12          Type 1    Builtin  yes yes no       6  0   UEWNID+SFBX1440        Type 1    Custom   yes yes no       7  0   FOVERL+SFRM1000        Type 1    Custom   yes yes no       8  0     latex+dvips+ps2pdf      name                                 type              encoding         emb sub uni object ID   ------------------------------------ ----------------- ---------------- --- --- --- ---------   WVMUEH+SFRM1000                      Type 1C           WinAnsi          yes yes no        16  0   ZHRBNN+SFBX1440                      Type 1C           WinAnsi          yes yes no      14  0   IMJWCD+CMMI12                        Type 1C           Custom           yes yes no      12  0   MDMSVR+SFRM1200                      Type 1C           WinAnsi          yes yes no      10  0   TWHNMT+SFRM1728                      Type 1C           WinAnsi          yes yes no       8  0 So, after a few sessions, the new player I'd invited has been a bit of a disaster. He doesn't get on well with any of the other players, laughed when someone's character died (in our campaign that's permanent at the moment), and generally has been a nuisance. However, he doesn't seem to notice this. I've talked to him about his behaviour, but it hasn't helped. Several other players have also approached me on their own to register concerns.  The long and short of it is, he's not a good fit for our game or social group, due to his abrasive personality/playstyle (as I and the other players perceive it). He's also not interested in roleplaying (something my group does heavily, at the moment) and continually pushes to place himself in the limelight - which he does by instigating combat and trying to skip over what he sees as 'boring' scenes. His character is a Minotaur barbarian, and the most exotic race in the game, but he only uses it for the stat bonuses, and doesn't even have a backstory - he simply says he doesn't need one. However, I don't want to be rude or otherwise unkind to him, as I will still see him around after he leaves the group, and so will the other players.  How can I politely ask him to leave the group, without being dishonest but avoiding embarrassment for anyone? I also want to avoid seeming to 'abandon' him, or 'go behind his back'.  Note: There are few/no in-game problems with adding or removing characters, and I don't need advice for that. This is strictly about handling the player. The system (not that it's all that relevant to the problem) is I need some help importing MrSID Raster files in QGIS. I am running 2.6.0 on Linux Mint 17. After an internet search I have found a couple of resources that are perhaps outdated. The most promising is found here :    However when I attempted to follow the instructions my terminal and I could not find the file "libgdal-mrsid-src" on either the stable or unstable Ubuntu GIS repos. Has anyone had success or could lead me in the right direction? Is drawing multiple cards an atomic action? Is drawing multiple cards an atomic action, or do effects triggered by drawing a card trigger for each card drawn instead?  Consider , which reads:     Whenever a player draws a card, Spiteful Visions deals 1 damage to that player.   If a player were to be compelled to draw more than one card, would that player be dealt the damage all at once, or one at a time? For example, how many times would  need to be activated to prevent all damage?  Furthermore, if there exist additional effects that are triggered by damage dealt in this way, are such effects resolved together atomically or is each resultant effect resolved individually?    How about tutors, or other cards not using the word 'draw', eg:      If you would draw a card, look at the top three cards of your library instead. Put one of those cards into your hand and the rest on the bottom of your library in any order. Problem with Winbind/Samba You can see the output from lscpu command -  jack@042:~$ lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                56 On-line CPU(s) list:   0-55 Thread(s) per core:    2 Core(s) per socket:    14 Socket(s):             2 NUMA node(s):          2 Vendor ID:             GenuineIntel CPU family:            6 Model:                 79 Model name:            Intel(R) Xeon(R) CPU E5-2690 v4 @ 2.60GHz Stepping:              1 CPU MHz:               2600.000 CPU max MHz:           2600.0000 CPU min MHz:           1200.0000 BogoMIPS:              5201.37 Virtualization:        VT-x Hypervisor vendor:     vertical Virtualization type:   full L1d cache:             32K L1i cache:             32K L2 cache:              256K L3 cache:              35840K NUMA node0 CPU(s):     0-13,28-41 NUMA node1 CPU(s):     14-27,42-55   I can see that there are 2 sockets (which is like a processor ??) and inside each of the socket we have 14 cores. So, in total 2x14=28 physical cores. Normally, a CPU can contain multiple cores, so number of CPUs can never be smaller than number of Cores. But, as shown in the output CPUs(s): 56 and this is what is confusing me.  I can see that Thread(s) per core:    2, so these 28 cores can behave like 2x28=56 logical cores.  Question 1: What does this CPUs(s): 56 denote? Does CPU(s) denote number of Virtual/Logical core, as it cannot be a Physical core core atleast?  Question 2: What does this NUMA node mean? Does it represent the socket? Microsoft.Office.Interop.Excel doesn't work on 64 bit How to run updmap-sys after installing cm-super? My original question was:   The answer that seems suitable to my cause is to install cm-super. I did that (using package manager [winxp 32 bit, MikTex installation 2.9 recently updated]). But still the command  \usepackage{cm-super}    yields an error.  User H Cavendish suggested to run updmap-sys, but I am not sure about how to do that on WinXP.   2011 November 04 edit: I made it sure that I have installed 'cm-super' - but still get the same error:   I have also tried updating the file name databases - but no avail. Restore Macintosh HD to its original partition configuration Are these plural or singular? I was surprised when I heard zero is plural, and even 1.00001 is plural.  Then, what about following numbers ?  (1) 1.00  which means the value measured  between 0.995-1.004, having possibility that it was exactly 1.  (2) 0.99999.....  which is mathematically equal to 1. Also, are others that are mathemattically equal to 1 than this, treated completely same as 1?  (3) -1  Is this singular? Or negative numbers are generally plural?  (4) i = square root of -1  I know it's weird to count things with imaginary numbers, nonetheless, if you were to count, which form, plural or singular, would you be to use?  That's it. What's the best solution to check a users country? What does "bad girls go backstage" mean? (sorry for my poor English)  I saw this phrase somewhere but I don't know what this means..     Good girls go to heaven, bad girls go backstage   What means 'bad girls go backstage'?   Maybe.. it's when good girls try to be good, but bad girls become famous?  Am I right? Adding a way to mark answer as "partially accepted" When someone gives me an answer which is useful, but doesn't fully answer the question, there is a problem when I need to decide what to do with it.  If I accept this partial answer, no one will answer me no matter how I show the unsolved part of my problem, because the question is considered "answered".  However, If I do not accept it, there is another problem:  if a second person answered the unsolved part, which answer should I accept? But the majority work may be done by the first one. If this style become consistent, who will answered an almost solved question?  In my point of view, there should be something like a progress bar which allows me to allocate the percentage of acceptance to users who answered my question. This would make everyone feel comfortable. In France the higher education is like this:   3 years of "Licence" 2 years of "Master" 3 years of "Doctorat"   But what about in the United States?  I heard that they have things called "undergraduate", "graduate" and "postgraduate". How many years do each of those take? And what are the equivalencies between these and the "Licence - Master - Doctorat"?  Note: If the answer to this question depends on the field of studies, then the field is physics. using tar showing, "No Such file or directory" But when I try to run it using  tar zxvf jre-8u151-linux-i586   I get tar (child): jre-8u151-linux-i586: Cannot open: No such file or directory tar (child): Error is not recoverable: exiting now tar: Child returned status 2  What is wrong? The name and place matches prefectly. Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &amp;\phi(x,a)=\lfloor x \rfloor-\sum_{p_i\le a}\left\lfloor \dfrac{ x }{(p_i)}\right\rfloor+\sum_{p_i&lt;p_j\le a}\left\lfloor\dfrac{ x}{(p_ip_j)}\right\rfloor-\sum_{p_i&lt;p_j&lt;p_k\le a}\left\lfloor \dfrac{x}{(p_ip_jp_k)}\right\rfloor+\dots \end{align}$$ then $\pi(x)=\phi\bigl(x,\pi (\sqrt{x})\bigr)+\pi (\sqrt{x})-1$ as given .  My question is, why does $\bigl\lfloor\phi(x,\lfloor\sqrt{x}\rfloor)+\sqrt{x}-1\bigr\rfloor$ also equal $\pi(x)?$ The external Flash is not showing up in the configuration Menu when using Pixel TF-321 hot shoe. Is this the wrong adapter for that camera, as there is no X connection? I want to trigger a studio flash via sync connection. Can you recommend me a working hot shoe? Thanks!