Let $\tau$ be the collection of sets $U \subset \Bbb R$ for which $U = \emptyset$ or $\Bbb R \setminus U$ is countable. Show that $\tau$ is a topology for $\Bbb R$.  Firstly I need to make sure that $\emptyset \in \tau$ and $\Bbb R \in \tau$. The first case is given in the problem statement, but I'm not sure if $\Bbb R \in \tau$ since I only have $\Bbb R$ with $U_i$'s subtracted? Also I would need to show that the arbitary unions of $U_i$, $i=1, \dots, n$ where $\Bbb R \setminus U_i$ is countable is in $\tau$. So in essnece $\bigcup_i(\Bbb R \setminus U_i)$ would have to be in $\tau$? How can I show this? I know that the unions of countable sets are countable so wouldn't that tell me immediately that $\bigcup_i(\Bbb R \setminus U_i)$ is countable? $\Bbb R$ clearly isn't, but is the difference?
I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology:     Let X be an infinite set. Show that $\mathscr{T}_2=\{U \subseteq X : U = \emptyset $ or $ X\setminus U $ is countable$ \} $ is a topology.   My book calls this set the ""  Please let me know if my proof is valid.     corollary [A]: The union of countable sets are countable.    proof: I'm going to take as a given that that the union of finite set are finite, and just prove the case of . A set is countable infinite if there is a bijection between the set and the natural numbers. Let $A$ and $B$ be a countable infinite sets. Now let $f$ be the bijection between $A$ and $\mathbb{N}$ and $g$ be $B$'s bijection.  $f:\mathbb{N}  \rightarrow  A$  $g:  \mathbb{N}  \rightarrow B$  There can always exist a bijection $h$ that maps $A \cup B $ to $\mathbb{N}$. To account for non disjoint sets, let $B' = B \setminus A$; (note: $A \cup B = A \cup B'$)  $h:  \mathbb{N}   \rightarrow  A \cup B'$  $ h(x) = \left\{         \begin{array}{ll}             f(\frac{1}{2}x); &amp; \quad x \in 2\mathbb{N} \\             g(\frac{x+1}{2}); &amp; \quad x \in 2\mathbb{N}+1         \end{array}     \right. $   $h^{-1}(x)$ would map $A$ to the even numbers, and $B'$ to the odd numbers.  Therefore, The union of countable sets are countable. (QED)  Now to verify the definition.     (i) X and Ø are elements of $\mathscr{T}$.   Ø is given in the definition of $\mathscr{T}_2$. Also, the empty set is considered finite, with cardinality zero. Thus $X$ is in $\mathscr{T}_1$ because its complement is finite, and finite numbers are countable.      (ii) $\mathscr{T}$ is closed under finite intersections.   Let:$V_k \subseteq X$ be any countable set. (ie: $V_1, V_2...$ are all countable sets)  So Then, $U_k = X \setminus V_k$, would be the subsets of $\mathscr{T}_2$.  $U_k \cap U_j = X \setminus (V_k \cup V_j) $, due to De Morgan's law.   Thus, $U_k \cap U_j \in  \mathscr{T}_2$, since the Union of two countable sets is also countable[A]. And This reasoning can be extended to any finite amount of intersections.       (iii) $\mathscr{T}$ is closed under arbitrary unions.   Let $A=\bigcup_{i \in I} U_i$ be an arbitrary union of sets whose complements are are countable.Then by De Morgan's law, $A=X \setminus (\bigcap_{i \in I}V_i)$, for some countable sets $V_i$. By the definition of intersection, $\bigcap_{i \in I}V_i \subseteq V_i$ for all $i \in I$. Since $V_i$ is countable so is any of its subsets.  Hence the complement of $A$ is countable.
In physics sometimes we find energy that is negative. What does the negative sign indicate? Inconsistency in display of animated avatars It has been discussed previously that animated avatars are disliked because they cause a distraction from the content. However, currently animated avatars are not treated consistently.   On desktop, animated avatars are displayed as a still image, both in posts and on the profile page. On mobile, animated avatars are displayed as a still image in posts, but animated on the profile page.   I'm tagging this  because I'm guessing one of these two is unintended. Either avatars should animate on the profile page on both desktop and mobile, or on neither. (I sincerely hope that we are all agreed there should be no animation of avatars in posts.)  Example animated profile    This appears to animate on mobile, and show a still image on desktop (for myself and for other users who have tried).    I'd also like to hear arguments for and against animating only on the profile page, but the main purpose of this post is to clear up the inconsistency. Installing Ubuntu on a SSD Going to install Ubuntu 10.10 on new intel x25M 80GB SSD. It will be fresh install. I have been googling for past few days and getting overwhelming articles/blogs/Q&amp;As. One particularly very useful being:     But with so many suggestions and differences of opinions (on different links) this simple OS install process seems to be daunting task to me and I really want to stick with Ubuntu (although have used for very short period of time).  Can someone help me by answering few questions (yes, they are repeated because I couldn't comprehend the answers elsewhere)   Which file system (ext2/3/4 or something else)? (consider SSD life) Can it be changed after installation? Should I partition the disk? (as we do in traditional HDD) for now, no plan of dual booting. Only Ubuntu will live on scarce space of 80GB SSD. I have 2 GB RAM, should i still allocate swap space (if I don't allocate swap space, can I still hibernate the machine)? will swap space impact SSD life? Should I consider putting additional 1GB RAM to avoid swap space? What is partition aligning? Is it needed to be done before installing the Ubuntu OS or can be done later?   Intended usage - heavy browsing, programming, regular video/music and some other non-CPU/RAM-intensive programs. Will keep big files in an external hard drive.  laptop config - 3yr old Vaio, Core2 Duo, 2GB RAM  More references:        (resizing/creating does not seem a simple process after look this) After removing any part the rest can be split evenly. Consequences? Let $S$ be a finite collection of real numbers (not necessarily distinct). If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same sum ; then is it true that all elements of $S$ are equal ?  (I know the result is true if the &quot;reals&quot; are replaced by &quot;integers&quot;.) How much to tip in the United States? What makes matter travel along geodesics? How to order files by last modified time in ruby? I want to install the latest Firefox on my Ubuntu (64bit) box.  What is the best way to do this? Is there a specific update site or should I download the binaries manually? If doing manually, in which directory should I put the files? In my home directory, or is there a better place? What is $\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$ for $c\in\mathbb{R}$ It is straightforward to show $$\int_{-\infty}^{\infty}\Phi(a+bx) \phi( x ) dx = \Phi\left(\frac{a}{\sqrt{1+b^2}}\right)$$ but what value does $$\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$$ have for $c&lt;\infty$, e.g., for $c=0$? Even a good approximation would really help me! I've observed several incidents where users inadvertently posted detailed, sensitive information. For example, a Server Fault user who posted a question with working login credentials to his actual production server, or the Stack Overflow user who posted actual medical records as example data for his question.   What should be the procedure for handling such cases? I often use  to keep my sentences gender neutral. Many of my friends (none of them are native speakers), however, consider this to be grammatically incorrect, and suggest me not to use it in formal English exams.  My questions are:   Is singular they widely accepted in English speaking countries such as America and Britain? Is it formal enough to be used in formal situations, such as a public speech? Is it casual enough to be used in informal situations, such as gossiping among friends? How to backup 20+TB of data? We have a NAS server at the company I work for that is being used for storing photography sessions. Each session is approximately 100gb. Over the last couple of years this server has accumulated 10+ TB of data, and we are increasing the amount of photoshoots exponentially. I estimate that by the end of next year we will have 20+ TB stored on this NAS. We are currently backing this server up to tape using LTO-5 tapes with Symantec BackupExec. Since the size of this server has grown, full backups of this server are not completing overnight. Does anyone have any suggestion on how to backup this amount of data? Should we be backing it up to tape? Are there any other options which may be better? Which words in a title should be capitalized? I was just wondering about that because it would influence how I deal with closed questions, e.g. whether they need to be improved (tagging, formatting, code samples) after closing, or if some should be deleted altogether.  However, I'm not asking for answers to these questions, I just want to know if Google (and/or other search engines) list closed questions in their search results. (I hope they don't.) X,Y,Z missing in F3 stop Ipod from asking for old Apple ID password How can i set different wallpapers for Ubuntu 15.04 login and Desktop This has been bothering me very much. i have memory that Ubuntu 12.04 had this configuration? am i wrong? finally, is it possible for the login screen to have different wallpaper from the desktop? is this possible in Ubuntu 15.04? i would appreciate a stable understandable response. thanks :)  -ben Should a retrieval method return 'null' or throw an exception when it can't produce the return value? Customise date-time format in Windows Is it possible to customize data (or date-time) format in Windows (I am using Windows&nbsp;XP)?  The current format which is followed by the OS [to show date-modified, etc.] is MM/DD/YYYY or M/D/YYYY, whereas I have been comfortable with DD/MM/YYYY or D/M/YYYY format. I am finding it hard to refer Date-modified [which I use often] of files and folders. Residue fields of schemes of finite type (over $\mathbb{Z}$) Suppose $X$ is a scheme of finite type over $\mathbb Z$. I want to prove that:  (1) The residue fields of closed points of $X$ are finite;  (2) For a given $q=p^n$ with $p$ prime, there is only a finite number of closed points of $X$ whose residue field is $\mathbb F_q$.  For (1), I see that one has to use some form of Nullstellensatz. First, we suppose $X=\operatorname{Spec}(A)$ affine, and $A=\mathbb{Z}[X_1,...,X_n]/I$. We need to show that if $m$ a maximal ideal of $A$, then $A/m$ is finite. Note $p=\mathbb Z\cap m$. If $p$ is maximal, we get that $A/m$ is of finite type over $\mathbb F_p$ thus a finite extension by Nullstellensatz and we are done. But $p$ could be the $(0)$ ideal in which case I neither conclude nor get a contradiction.  For (2), we can again argue locally, so we need to show that $A$ contains only a finite number of maximal ideals $m$ with $A/m$ of fixed cardinality $q=p^n$. I get an impression that to count these ideals its the same as to count the number of automorphisms of $\mathbb{F}_{q}$ which fix $\mathbb F_p$, but the corrspondence is not bijective, so we only get an upper bound which is enough. Is this idea correct?