I got this question in my topology exam and I had no idea how to make it, topology can be so weird sometimes it's hard to imagine some spaces.  My heart told me that it is not possible, but I do not know how to proceed on this. The easiest way seems to be using some topological invariant, line connectedness.   Please note that the factors on the product are the same.
I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism?
Control Android with broken screen from PC My rooted Galaxy Nexus has a broken screen. How can I use my PC (ie, via ADB) to control the phone?  androidscreencast would be the ideal solution, but its keyboard/mouse control feature does not work in Jelly Bean. Perhaps someone knows why and we can patch the code? (see: )  Installing a VNC server is another, but for the love of me I can't get it to work. Eg, using Droid VNC Server, I install the apk, and am supposed to execute /data/data/org.onaips.vnc/files/androidvncserver but there is no files directory inside org.onaips.vnc (instructions from here: ). VMLite is another VNC server that can be started with a special Windows utility (and doesn't need root!) but it's not OSS so I can't get the apk in order to install (Google Play is not pushing apps to my phone prob because I disabled some service--let me know if I can force Google Play to update itself from ADB or install the app). fastdroid-vnc is a binary (not apk) and it runs fine via shell, but something is broken and it shows nothing.  What are my options? Let's make this a community wiki, to help anyone who's broken their screen! How do I create a reflective white material? I'm pretty new to this and this is my first question here, any help would be greatly appreciated.  So basically what I want to achieve is a floor material as seen in this photo...    ...the reflections are very crisp but dim.  With a glossy shader I get the following results:    First picture: reflections are blurry  Second picture: roughness of the material looks right but reflection is bright red  Third picture: If tinted in any other color than white, it seems to produce the result I want. Reflection is sharp and gets tinted blue. What I need is the reflection to be tinted white.  Thanks for any help. I am installing Ubuntu 16.04 in an Intel NUC computer with UEFI enabled. I am using an SSD from another computer. During the installation I chose Erase the disk and make a new full install  At the end of the process I see:  grub-efi-amd64-signed failed installation /target/ Ubuntu 16.04    And the system does not boot.  I tried to use boot-repair but apparently it does not solve the problem. How do you correctly say large numbers I saw a post on  which links to a video where a person . Is he saying a large portion of the numbers wrong? Back in high school my algebra teacher was extremely picky.  If we were going to say the number 135, we would say one hundred thirty-five.  If a person said one hundred and thirty-five my teacher would interpret that to mean 100.35.  Was my algebra teacher wrong, or did Jon just waste over 80 hours counting to 100,000 incorrectly. I tried searching the web, but searching for ‘numbers and’  doesn’t work very well.  Can you point me to a reference or style guide that describes how to properly say large numbers? Passing functions to setTimeout in a loop: always the last value? I have a battery pack of NiMH batteries. It is ten cells with 1.2V, 4000mAh each, put together in series. So rated voltage is 12V. After charging, i.e. when the charging device says that it is finished, I measure 14.3V.   When I now discharge, at what voltage should I consider my battery as "empty"? Is this the lowest voltage that I can discharge it to without damaging it? Which voltage is that? Can I use variables inside {} expansion without `eval`? Can I use variables inside {} expansion without appealing to eval? If so, how?  This doesn't work:  $ touch 1.foo 1.bar $ ls 1.{foo,bar} 1.bar  1.foo $ extensions=foo,bar $ ls 1.{$extensions} ls: cannot access 1.{foo,bar}: No such file or directory   It works with eval:  $ eval ls 1.{$extensions} 1.bar  1.foo Share Wireless connection with Wired Ethernet Port Zsh says "no matches found" when trying to download video with youtube-dl I downloaded youtube-dl using pip on CrunchBang (a Debian Wheezy distro). When I run   youtube-dl {video URL}   I get  zsh: no matches found: {video URL}   I guess it has something to do with zsh, because I found some users on Arch forums complaining about it. When I switched to bash it worked. Possible Duplicate:          There are a lot of really screwed up configurations out there, because people didn't take a minute to sit down and think about something as simple as the name for their Active Directory forest. I'm asking this question to self-answer so that this knowledge can be more readily available. If you think you have a better answer, by all means add your input (but be prepared to defend it!)   So, what should I name my new Active Directory? How can I redirect the user from one page to another using jQuery or pure JavaScript? Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.   This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.   (Please note that $e$ in the question is the group's identity.)  Here's my attempt though...  First I understand Abelian means that if $g_1$ and $g_2$ are elements of a group $G$ then they are Abelian if $g_1g_2=g_2g_1$...  So, I begin by trying to play around with the elements of the group based on their definition...  $$(g_2g_1)^r=e$$  $$(g_2g_1g_2g_2^{-1})^r=e$$ $$(g_2g_1g_2g_2^{-1}g_2g_1g_2g_2^{-1}...g_2g_1g_2g_2^{-1})=e$$  I assume that the $g_2^{-1}$'s and the $g_2$'s cancel out so that we end up with something like,  $$g_2(g_1g_2)^rg_2^{-1}=e$$ $$g_2^{-1}g_2(g_1g_2)^r=g_2^{-1}g_2$$  Then ultimately...  $$g_1g_2=e$$  I figure this is the answer. But I'm not totally sure. I always feel like I do too much in the pursuit of an answer when there's a simpler way.  Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra. KOMA-Classes (scrbook, scrreprt, ... ): Disable chapter separation in \listoffigures and \listoftables I use \listoffigures and \listoftables. Both commands create line breaks for items from different chapters. How can you stop that?  Screenshot: Here is my situation: I know almost nothing about Perl but it is the only language available on a porting machine. I only have permissions to write in my local work area and not the Perl install location. I need to use the  Perl module from CPAN   How do I use this Parallel::ForkManager without doing a central install? Is there an environment variable that I can set so it is located?  Thanks  JD How much Valyrian steel is there that is unused for link making and why aren't people making weapons out of it? I was reading  that stated Maesters can have Valyrian steel links:     Valyrian Steel links (for the study of Magic + the Occult) are exceedingly rare and held by less than one Maester in a hundred. Ironically, they aren't seen as especially prestigious but rather viewed with suspicion.   This made me wonder how much Valyrian steel there is lying around for Maesters to be adding them to their chains. It seems like such  resource like this would be used for crafting weapons not being kept as ingots ready for Maesters to make links.  Indeed the Wiki I linked above claims:     the amount of Valyrian steel in the world is finite and extremely rare   How much of his extremely rare resource is being kept in reserve for link making and why hasn't it been used for more useful stuff like making weapons? I am curious as to why "nay" replaces the simple unequivocal "no" in the context of voting.  My research in Merriam-Webster tells me that "nay" means "no" (not the other way around) and the first known use of "nay" appeared in 1400s. Since the use of "no" existed before 1400 and the word remains to be used afterward, why would "nay" become the preferred choice of word when it comes to voting. Any idea? How (and why) use stderr for both reading and writing? According to , less reads navigation commands from stderr if it's not able to open /dev/tty.  This seems puzzling, since I've never seen anything write to another program's stderr stream, and I don't know how I would even accomplish that.  What is the purpose of stderr being open for both reading and writing? And if this is useful, how do I make use of it on modern systems? (Is there some arcane syntax to pipe something into stderr instead of stdin, for instance?) Will i be granted ESTA after getting rejected thrice on different nationality? I had applied for B1/B2 on my previous nationality but it got rejected Thrice.  I had planned on visiting my mother as she lives there. my last refusal was 3 years back. Now i have gained an European nationality and I am thinking of applying for ESTA so I can visit my mother. I have doubts if it will be accepted because I'll have to disclose about my previous refusals.  What should I do? because if I don't disclose my previous refusals I fear I might be rejected permanently. Does ClamAV Scan Windows virus? I'm new to Ubuntu.  I'm now setting up an OwnCloud server which supports ClamAV file scan.  Since most of the users connected to the server uses Windows OS, can ClamAV scan for Windows virus?  I also did some research that such as BitDefender. It clams that it scans real time, if someone upload files to the server, will BitDefender scan the file even OwnCloud doesn't support?  By the way, I heard that if I update the system frequently, I don't have to install any antivirus that scan for Linux virus. Is that right? Why can't this be done with Master Theorem Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem?