See, I am told that $(0,1)$ is not compact as a subspace of $\mathbb{R}$. Question is, how do I conclude that?  The hint says      $(\epsilon,1)_{\epsilon&gt;0}$ does not have a finite subcover.   But I am unsure why $(\epsilon,1)$ does not have a finite (open) subcover for $(0,1)$. The right end point is fixed to $1$ so all I can do is move around $\epsilon$. So some $\epsilon$ as close as possible to zero $(\epsilon,1)$ where $\epsilon \to 0$ alone would be an open cover of $(0,1)$.  Any other $\epsilon&gt;0$ I choose would be redundant. $(1/2,1),(1/60,1),(8/7,1)...$ whatever. These are already covered by $(\epsilon,1)$ where $\epsilon \to 1$. So that means, the open cover by $(\epsilon,1)$ is finite on its own to start with. namely, just one open subset $(\epsilon,1)$ of $\mathbb{R}$ covers $(0,1)$.  I can come up with loads of other open covers, sure, but as I said, any other open cover apart from $\epsilon \to 0$ can be removed and still be an open cover of $(0,1)$. e.g. $\{(\epsilon,1)_{\epsilon \to 0},(23/81,1)\}$ is an open cover of $(0,1)$ but I can get rid of $(23/81)$ if I want to since that still leaves me with an open cover of $(0,1)$. So the same for any infinite open cover $\{(\epsilon,1)_{\epsilon \to 1},(1/2,1),(2/5,1),(1/3,1),...\}$ I can get rid of everything that follows $(\epsilon,1)$ and be left with an open cover i.e. finite subcover.    I need some clarification here and most probably; I am mistaken with something so blatantly obvious to those with experience. Think I am a super stupid human being and explain from possible the $1+1=2$s of this bizarre world of compactness if necessary. I think someone needs to fix some basic but esoteric notion for this one.
Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover.  What is wrong with saying that, in $\mathbb{R}$, if $I=(0,1)$, we can choose $G=\{(0,\frac{3}{4}), (\frac{1}{4}, 1)\}$, which satisfies $I \subset \bigcup_{U\in G} U$, but we can't extract a finite subcover, so $I$ is not compact. Is $G$ a finite subcover of $G$, so it is not a valid cover for proving this? I would take $\cup_{n\in\mathbb{N}} (\frac{1}{n},1)$ in order to prove this, can we conclude that every open cover is necessarily a infinite union of open sets $\neq \emptyset$?
The outlet for my range hood was misplaced. It needs to be 4 inches to the side. I can make the junction and move the wiring to the new location but I really don't want to have the empty plate exposed.   The wire is not long enough to reach the required location, so I will make a junction in the existing box, and add a new box at the correct location.  Is there any acceptable circumstance where I could tile over the "old" in-wall junction box?  I'm guessing not, but I figured I'd ask.  EDIT: I should clarify, based on the answers below... I know for a fact that this outlet is the only line on this entire circuit, all the way back to the panel. I am asking explicitly if there is any way to acceptably splice the wiring without exposing a junction box. How can I switch between starting a coroutine vs running the routine all at once? How to install Wi-Fi driver for Realtek RTL8821CE on Ubuntu 18.04? I'm trying to install drivers for my Wifi on my HP All-in-one desktop. I've been told to show the output of the command "sudo lshw -C network" so here it is:  *-network description: Ethernet interface product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller vendor: Realtek Semiconductor Co., Ltd physical id: 0 bus info: pci@0000:01:00.0 logical name: enp1s0 version: 15 serial: 48:ba:4e:5c:06:5f size: 100Mbit/s capacity: 1Gbit/s width: 64 bits clock: 33MHz capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168h-2_0.0.2 02/26/15 ip=192.168.1.21 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s resources: ire:31 import:e000(size=256) memory: fea04000-fea04fff memory:fea00000-fea03fff  *-network UNCLAIMED description: Network controller product: Realtek Semiconductor Co., Ltd. vendor: Realtek Semiconductor Co., Ltd. physical id: 0 bus info: pci@0000:02:00.0 version: 00 width: 64 bits clock: 33MHz capabilities: pm msi pciexpress cap_list configuration: latency=0 resources: import:d000(size=256) memory:fe900000-fe90ffff   Output from "lspci -nnk | grep -A2 0280":  02:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821CE 802.11ac PCIe Wireless Network Adapter [10ec:c821] Subsystem: Hewlett-Packard Company RTL8821CE 802.11ac PCIe Wireless Network Adapter [103c:831a] 03:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. RTS5229 PCI Express Card Reader [10ec:5229] (rev 01) Change the Ubuntu program that runs when a command can't be found I know mathematically the answer to this question is yes, and it's very obvious to see that the dimensions of a ratio cancel out, leaving behind a mathematically dimensionless quantity. However, I've been writing a  (the specifics of which are out of scope), which has me thinking about the problem because I decided to handle angle units as dimensioned quantities, which seemed natural to enable the unit conversion with degrees. The overall purpose of the library is to disallow operations that don't make sense because they violate the rules of dimensional analysis, e.g. adding a length quantity to an area quantity, and thus provide some built-in sanity checking to the computation. Treating radians as units made sense because of some of the properties that dimensioned quantities seemed to me to have:  The sum and difference of two quantities with the same dimension have the same physical meaning as both quantities separately. Quantities with the same dimension are meaningfully comparable to each other, and not meaningfully comparable (directly) to quantities with different dimensions. Dimensions may have different units that are scalar multiple (sometimes with a datum shift).  If the angle is treated as a dimension, my 3 made up properties are satisfied, and everything &quot;makes sense&quot; to me. I can't help thinking that the fact that radians are a ratio of lengths (SI defines them as m/m) is actually critically important, even though the length is cancelled out. For example, though radians and steradians are both dimensionless, it would be a logical error to take their sum. I also can't see how a ratio of something like (kg/kg) could be described as an &quot;angle&quot;. This seems to imply to me that not all dimensionless units are compatible, which seems analogous to how units with different dimensions are not compatible. And if not all dimensionless units are compatible, then the dimensionless &quot;dimension&quot; would violate made-up property #1 and cause me a lot of confusion. However, treating radians as having dimension also has a lot of issues, because now your trig functions have to be written in terms of $\cos(\text{angleUnit}) = \text{dimensionless unit}$ even though they are analytic functions (although I'm not convinced that's bad). Small-angle assumptions in this scheme would be defined as performing implicit unit conversions, which is logical given our trig function definitions but incompatible with how many functions are defined, especially since many authors neglect to mention they are making those assumptions. So I guess my question is: are all dimensionless quantities, but specifically angle quantities, really compatible with all other dimensionless quantities? And if not, don't they actually have dimension or at least the properties of dimension? I recently moved from having a main site like www.example.com with a subdomain associated site yyy.example.com, to moving everything off the subdomain to a subdirectory like www.example.com/yyy/, with the same hierarchy underneath the subdirectory, and now yyy.example.com is just a redirect page to the subdirectory, so yyy.example.com/abc/page-x is redirected to www.example.com/yyy/abc/page-x and so on.  The effect has been that the traffic increased by several times (the total page views are more than ten times at the present rate) what it was to the yyy subdomain. Previously almost all of the traffic to the yyy subdomain was from the www site. This is from monitoring the old yyy site for a year or more.  I'm curious to know if there's any research or results which would indicate whether this is universal (subdirectories beat subdomains) or "just me". Cocos2d-x supporting multi-resolution using density independent pixels Is there a way to use Density Independent pixels(dp) in cocos2d-x?  I am developing an Android game using Cocos2d-x 3.0.  The Android reference  tells me not to use pixels in code but use dp when supporting multi-resolution support. It also tells me to provide different images for different screen size groups and different density groups.  However, cocos2d-x tutorials use pixels for multi-resolution support. If I were to use pixels, I think it means I need to have images for all these different resolutions:  240x320, 240x400, 240x432 320x480 480x640, 480x800, 480x854 600x1024 640x960 1024x600, 1024x768 1280x768, 1280x800 1536x1152 1920x1152, 1920x1200 2048x1536 2560x1536, 2560x1600  Even if I don't support every screen resolutions, I think this requires more work than following the Android reference's best practice.  Am I missing something on how supporting multi-resolution works or do I indeed do need to do much more work using cocos2d-x? Where can I find historical data on flights, specifically Delta international flights?  I need to find out about cancellations, diversions and any other flight data pertaining on particular route at a given time.  This source has to be official and stand up to legal scrutiny. What are alternatives to ExtJS? So what I'm looking for is a javascript framework I can use that has several UI controls. I have taken a look at  but those controls are very basic compared to ExtJS. Are there any other competitive alternatives?     This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. More info: . launch app, capture stdout and stderr in c++ Does anyone know how to model a depletion mode N channel MOSFET in spice?  I have looked online however the majority of resources have shown how to model a depletion type or enhancement type. I specifically want to model a FET that is fully on when there is a 0 Vgs voltage.  Resources i've looked at Solving the matrix equation $AP=PB$, where $A$ and $B$ are similar Currently I'm stuck with this problem:     The following matrices are similar to each other.      $$A=\begin{bmatrix}-3&amp;0\\8&amp;4\end{bmatrix},\qquad B=\begin{bmatrix}-4&amp;4\\-2&amp;5\end{bmatrix},\qquad C=\begin{bmatrix}k&amp;1\\m&amp;-2\end{bmatrix}.$$      How do you find the matrix $P$ such that $AP = PB$?       And how can you calculate the values of $k$ and $m$?      Thank you in advance. Why is it called an “Indian file”? Finding CPU usage of a service Because my laptop fan was spinning up for no reason, I had a look at the CPU usage and I see  svchost with 12.5% CPU, which is basically a full core.  Now, I'm aware that  svchost runs services and I know how to display the services it runs in Task Manager or Process Explorer ().  Now I know which services are run, but I don't know which service eats the CPU time. I could disable one service after another, but I don't really like doing this (it could break things that are currently in use and I could forget turning the service back on later).  Therefore I'd like to know whether it's possible to instruct Windows to run only one service per svchost the next time I boot my PC. While I might have the slight disadvantage of several processes, I could immediately find out which one it is and it would be a much easier decision of whether or not to stop or restart the service. Linux command line. Move all files and directories in directory, except some files and directories When you click on the Opportunities tab, (assuming you haven't modified standard salesforce layout), you see a "Recent Opportunities" list. At the top of the list, there is a "New" button.     Is there a way to hide the "New" button displayed on this page? I have tried disabling the button on the "Opportunity List View" Search Layout[Setup -> App Setup -> Opportunities -> Search Layouts], but it does not remove the button from this particular view. LaTeX Error: Option clash for package xcolor I have an option clash between package xcolor and another package, but I don't know which. Here is my MWE  \documentclass[10pt]{beamer} \setbeamertemplate{navigation symbols}{} \usepackage{caption} \usepackage{subcaption} \usepackage{url} \usepackage{color} \usepackage[usenames]{xcolor}  \author{The author} \title{Package conflicts} \date{October 3rd, 2012}  \begin{document}  This is supposed to be \textcolor{olivegreen}{olivegreen} \end{document}   And the error message is :  This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./error_xcolor.tex LaTeX2e &lt;2009/09/24&gt; &lt;blablabla&gt; ! LaTeX Error: Option clash for package xcolor.   I hope this is enough log for you guys. At this time of the day, I'm hoping for America to save my french ass ... again : ) Ubuntu 13.10 Macbook Air Mavericks Parallels 9 Black Screen I can install Ubuntu 13.10 in parallels ok and then log in.  Problem is Ubuntu does an update and I get black screen / desktop such that I can't see any app windows that I open.  I've tried a few of the other suggestions and they don't work for me. How to use Whatsapp mobile application on Ubuntu Touch? In my Android I have "Whatsapp" application running. But mobile freak I am not, and specially the mobile keyboards.  So how can I run my same application, same account from my Ubuntu desktop PC?   Follow up:  Just to do keyboard routing, i have used . Which allows me to takeover the Android input fields access via web link to type in my PC keyboard, and then it re-route the keys to Mobile.  Features:   Allows me to copy and paste large texts to Android input fields without using the keyboard of Android but simply using my Desktop PC, via web link http://android.ip.address.goes.here:port/ Allows me to have key submit which is similar to key press of mobile set itself   At-least works without having mobile in rooted. Hope it helps you!! How do you find out the caller function in JavaScript? function main() {    Hello(); }  function Hello() {   // How do you find out the caller function is 'main'? }   Is there a way to find out the call stack?