Can an open set in $\mathbb{R}^n$ homeomorphic to a non-open set? Suppose $A$ is an open set of $\mathbb{R}^n$ under the ordinary topology and is homeomorphic to $B\subset \mathbb{R}^n$. Is it necessary that $B$ be open too?
Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic? Suppose $A$ is an open subset of $\mathbb{R}^n$ considered as a topological subspace. Suppose $B$ is a another susbet of $\mathbb{R}^n$, and under the induced topology, is homeomorphic to $A$.  If $B$ is closed and bounded, then by the Heine-Borel theorem, $B$ is compact as a topological space so $A$ and $B$ cannot be homeomorhpic, because $A$ is not compact. But if $B$ is not bounded, what can be said about $B$, if any? Must $B$ be open, can it ever be closed (let's leave out the only case of being both open and closed--the entire ambient space $\mathbb{R}^n$), or can it be neither open nor closed?
Korean characters scrambled [XeLaTeX; Noto Sans Korean]? I was trying to find a font which has obsolete jamo and the right combining behaviour. Yesterday I found out about , which seems to have both of these. However, when I use it in XeTeX, I get some seemingly-arbitrary hanja (or blank spaces) instead of the expected characters.  (I’m guessing the composing behaviour will fix itself when the right characters are selected. And yes, I know that Code2000 is working perfectly in this example, but unfortunately it is the ugliest font in the world. :p)  MWE:  \documentclass{article} \usepackage{fontspec}  % 1. precomposed: U+BB47 % 2. separate jamo: U+1106, U+116E, U+11BA % 3. obsolete [separate jamo]: U+113B, U+119F, U+11EB \newcommand\test[1]{#1 &amp; {\fontspec{#1}{뭇; 뭇; ᄻᆟᇫ}} \\}  \begin{document}  \begin{tabular}{rl} \test{Arial Unicode MS} \test{Batang} \test{Code2000} \test{Gulim} \test{GungSeo} \test{HeadLineA} \test{Nanum Brush Script} \test{Nanum Myeongjo} \test{Nanum Pen Script} \test{Noto Sans Korean} \test{PCMyungjo} \test{PilGi} \test{Adobe Gothic Std} \test{Adobe Myungjo Std} \test{Apple SD Gothic Neo} \test{AppleGothic} \test{AppleMyungjo} \end{tabular}  \end{document} I've got a development webserver on an Ubuntu box inside of VMWare Workstation 7.1.3. I'd like to run a web browser on a Windows machine (the host OS) for actually testing a web page I'm working on -- hard to run IE tests in Ubuntu :)  However, if I simply try to ping the IP given to the ubuntu VM (192.168.83.137) (or open it in my browser for that matter), it seems the host cannot communicate with the guest.  How do I allow the guest OS to communicate with the host when the guest is running in NAT mode? Simple form validation Why is $H(k\mathbin\Vert x)$ not a secure MAC construction? How closely can a game legally resemble another? How to change this matrix with respect to standard basis? Given the basis $\beta = \{ (1, 1, 0),\ (1, 0, -1),\ (2, 1, 0)\}$ and the matrix:  $$ A =         \begin{bmatrix}         1 &amp; 1 &amp; 0 \\         1 &amp; 1 &amp; 0 \\         0 &amp; 0 &amp; 1 \\         \end{bmatrix} $$  with respect to  $\beta$.  How to change this matrix with respect to standard basis? Badge Notification Repeats Make an object invisible to a specific camera? (Cycles) Prevent pdflatex from writing a bunch of files If I create just a basic hello.tex file an compile it using   pdflatex hello.tex   The pdflatex program is creating a plethora of file which is a little annoying. Currently it writes all those files about which I really don't care:  hello.aux, hello.log, hello.nav, hello.out, hello.snm, hello.toc   Is there a way to tell pdflatex to only write hello.pdf if the build is successful ? If not, can I specify the output directory of those files ? Maybe I could just write them in /tmp. Every time I try to create a table using MySQL Workbench 6.3 CE it adds "broken" comments at the end of the script.  Example:  I am trying to create a table called "user". This is what the table should look like:   However when I click the apply button, the script generates incorrectly:  CREATE TABLE `user` (   `userid` INT NOT NULL AUTO_INCREMENT COMMENT '',   `record_status` INT NOT NULL DEFAULT 1 COMMENT '',   `createdby` VARCHAR(75) NOT NULL COMMENT '',   `createddate` DATETIME NOT NULL COMMENT '',   `updatedby` VARCHAR(75) NULL COMMENT '',   `updateddate` DATETIME NULL COMMENT '',   `username` INT NOT NULL COMMENT '',   `password` VARCHAR(75) NULL COMMENT '',   `firstname` VARCHAR(75) NULL COMMENT '',   `lastname` VARCHAR(75) NULL COMMENT '',   `email` VARCHAR(75) NULL COMMENT '',   `facebook_userid` VARCHAR(75) NULL COMMENT '',   `email_verified` VARCHAR(75) NULL COMMENT '',   PRIMARY KEY (`userid`, `username`)  COMMENT '');   The comment in this line should not be there, so the code should be:   PRIMARY KEY (`userid`, `username`));   and the code executes when I paste it into an editor and remove this comment. However I create massive databases daily. (Each with +- 15 tables). I can't keep doing this for each table as it is time consuming.  How can I disable this feature that adds the comments to the script? I don't need them in my databases ListPlot and filling between curves What's a better way to define individual link colours for a url in a macro? I've got a document where I'd like two colours of links - cyan for links in headers that since it's brighter, and navy blue for links in the main body which is more subtle/less distracting. I already tried using the xcolor package and changing the text colour with textcolor, like \textcolor{cyan}{\href{url.com}{link name}}, but it doesn't override whatever link or url colour I have defined in hypersetup.   So this is what I've done:   I've defined the colour navy:   \definecolor{navy}{HTML}{2F729C}    And also imported the hyperref package and defined commands to set the global link/url colours to cyan / navy.   \usepackage{hyperref}                         \hypersetup{colorlinks=true,linkcolor=navy,urlcolor=navy}                        \newcommand{\navylinks}{\hypersetup{linkcolor=navy,urlcolor=navy}}               \newcommand{\bluelinks}{\hypersetup{linkcolor=cyan,urlcolor=cyan}}     Then I've defined another command specifically for the header links, where I turn on cyan links globally, include the header text and link, and then turn on navy links again:   \newcommand{\headerlink}[1]{\bluelinks\href{#2}{#1}\navylinks}   My issue with this is that I feel like there could be a better way to do it than to set the global link colour every time I use a link macro; is there a way to define the link colour just for one link, like one would do with \textcolor{...}, that would be able to override hypersetup? nginx httpMapModule regex variables The  states that:     Since 1.0.4, case insensitive regular expressions can be used by prefixing the pattern with ~*.   And gives the example:  map $uri $myvalue {     /aa                   /mapped_aa;     ~^/aa/(?&lt;suffix&gt;.*)$  $suffix; }   I updated to nginx/1.2.7 to make use of this, and naive regexes work okay, but variables don't. When I try the above syntax I get the error:  nginx: [emerg] pcre_compile() failed: unrecognized character after (?&lt; in "^/aa/(?&lt;suffix&gt;.*)$" at "suffix&gt;.*)$"   I also tried the following syntaxes, but none of them worked:  ~^(.*)$  $1; ~^(.*)$  \1;   but neither of them worked either.  Does anyone know how to get variables to work in regular expressions in HttpMapModule? Learning Insertion Sort in Ruby I have just started the MIT Introduction to Algorithms course through the material posted online. Along with the course I have also decided to learn/enhance my Ruby skills by coding the algorithms in it.   I am on the first algorithm given, which is Insertion sort, and I have the following code typed up but I am getting this error when I run it:     insertionsort.rb:5:in `>': comparison of Fixnum with nil failed (ArgumentError)   def  insertionsort(num) for j in 2..num.length     key = num[j]     i = j - 1     while i &gt; 0 and num[i] &gt; key         num[i+1] = num[i]         i = i - 1     end     num[i+1] = key end  puts num end  numbers = [23,34,46,87,12,1,66]  insertionsort(numbers)   I'm sure it is a fairly basic problem but I just can't grasp what it is at the moment. Any  help or tips would be very much appreciated. Is there an injective function $f :\mathbb{R}^2\rightarrow\mathbb{R}$?  I approached this problem from the perspective of $f^{-1}$, from which I showed there exists a surjective function $f^{-1}:\mathbb{R}\rightarrow\mathbb{R}^2$. Would this imply that there exists an injective function for the inverse mapping? I have Windows 8 Pro activated with kms. when I open windows store, it doesn't show "win8.1 available". clicking on ms-windows-store:WindowsUpgrade just brings me to windows store main screen  slmgr /dli shows volume_kmsclient channel  I can upgrade it to WMC. Laptop is W8Pro WMC and update is available, it's retail channel  Is W8 for KMS clients delayed or what? If $A$ is an $m \times n$ and $B$ is an $n \times m$ matrix with $n &lt; m$ then prove that $C = AB$ is not invertible.  I am not getting any idea. May be I am missing vary basic point. $C = AB$ where both $A$ and $B$ are square matrices  will be invertible iff both $A$ and $B$ be invertible. I know it and its proof.  May be this question has been discussed earlier, but I am not seeing the link. Thank you for your help. Dual boot Win 8 / Ubuntu loads only Win My computer is a Sony Vaio Notebook. I used Ubuntu and Win 8 with dualboot but windows somehow managed to remove the menu to choose the OS at startup. I think the BIOS is UEFI. Secure boot was disabled already and Boot Repair somehow fails.  How can I fix this? Can a curve object modifier created geometry be used as boolean factor This question is related to .  Let me use an exemple to illustrate what I mean.  We've created a text object (made of curves). It can't be used in a boolean modifier as it won't appear in the list of usable objects in the modifier because it's a curve object.  But if we add a remesh modifier on the text, geometry is created from the shape.  The object could be used as boolean if we convert it to mesh. The remesh can't be applied on curve object.  My question is : Can we dynamicaly use the remesh modifier created geometry as real geometry or make the boolean modifier consider the text modified object as a real mesh?  The idea behind this is to be able to change the text (or any curve 3D shape) without having to make it a real mesh.  (don't hesitate to edit my poor english to make it understandable :) ) How to install the latest Arduino IDE?