Can an open set in $\mathbb{R}^n$ homeomorphic to a non-open set?
Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic?
Mask and inherit the same systemd unit Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? Root cause for "Entity is not org-accessible" for sharing object? elementary ring proof with composition of functions and addition of functions as operations Consider the set $\mathcal{F}=\lbrace f\mid \mathcal{f}:\Bbb{R}\to\Bbb{R}\rbrace$, in which an additive group is defined by addition of functions and a second operation defined as composition of functions. The question asks to verify that the resulting structure does not satisfy the ring properties.     This is the question 24.10 from Modern Algebra, by Durbin, 4th edition.  So what I have so far is that all of the properties for the additive group (with addition of functions as the operation here) hold, associativity of the composition of functions hold, and that the failure must come from the distributive laws.  My proof of my claim, so far;  $$1\,\,\,\,\,[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x})=\cdots=\mathcal{f}(\mathcal{g}(\mathcal{x}))+\mathcal{f}(\mathcal{h}(\mathcal{x})).$$ ...but,  $$2\,\,[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x})=$$ $$3\,\,(\mathcal{f}\circ\mathcal{g})(\mathcal{x})+\mathcal{h}(\mathcal{x})=$$ $$4\,\,\,\mathcal{f}(\mathcal{g}(\mathcal{x}))+\mathcal{h}(\mathcal{x}).$$ ...which brings me to my contradiction of the properties of rings that;  $$5\,\,\,[\mathcal{f}\circ(\mathcal{g}+\mathcal{h})](\mathcal{x})\neq[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x}).$$  So my question is, is this the correct path to showing the distributive laws are not obeyed when composition is taken as an operation on $\mathcal{F}$? particularly my proceeding from 2 to 3 to 4 is what i feel uncomfortable about, like i may be missing a step in there.  thanks Moon photography Hi I am a photography hoobyist. At the moment I am using a Nikon D3100 with the standard kit lens (18-55mm). I am interested in taking photos of the moon. I am debating whether to buy a 55-200mm lens or a 55-300mm lens. Which one would be better?  P.S I am on a tight budget and I am looking for a lens for personal use, not commercial.  Would appreciate any answers!  Thank you! I have equations inside an environment which has a larger left-margin than the main text (analogous to an enumerate/itemize environment, implemented with the changepage package's adjustwidth environment). Just as equations in enumerate/itemize behave, the equations are centered with respect to the entire text, rather than just the width of the container environment. I want the latter behavior.  I'm trying to accomplish this by wrapping all display-math in minipages. Namely, I'm modifying the mathdisplay environment in amsmath.sty as follows:  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother   This is almost working, but there are some issues with spacing which I don't understand how to fix. Here's how the the normal \[ ... \] and numbered \begin{equation} ... \end{equation} look with the above modification:      My main confusion is why the numbered version is behaving so differently from the unnumbered. But all in all I want to fix this so that the spacing above and below is even as it usually is. Any help is appreciated, thanks.  Edit: Here's a compilable example:  \documentclass{amsart}  \usepackage{lipsum} \usepackage{calc} \usepackage{changepage}  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother  \setlength{\parindent}{0pt}  \begin{document}  \lipsum[1]  \begin{adjustwidth}{20pt}{} \lipsum[2] \[ x \mapsto \hom(-,x) \]  \lipsum[3] \begin{equation}   y \mapsto \hom(y,-) \end{equation} \lipsum[4] \end{adjustwidth}  \lipsum[5]  \end{document} Defining starred versions of commands (* macro) Pre-image of a measurable set A is always measurable? Let $f$ be a continuous function on a set $E$.  Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable?  I say no.  We know that $\bar{\psi(x)}:=\frac{\phi(x)+x}{2}$ where $\phi$ is the Cantor (or Cantor-Lebesgue) function and $\bar{\psi}:[0,1] \rightarrow [0,1]$.  We know this function maps a measurable subset of $C$ (the Cantor set) into a non-measurable set $W$.  A few observations about $\bar{\psi(x)}$.  It is strictly increasing and continuous, so it has a continuous inverse $\bar{\psi(x)}^{-1}$.  Thus $\bar{\psi(x)}^{-1}(W)=c \subset C$; that is, $\bar{\psi(x)}^{-1}(W)$ is non-measurable, but c is.  Is this correct? Do airlines have to check the visa status before selling a ticket? How does the momentum operator act on state kets? How to set systemwide "XP style DPI scaling" in Windows 8.1? I run my ThinkPad with a system DPI setting of 140%. In Windows 7 and 8, I enabled the "Use Windows XP style DPI scaling" checkbox in the Custom DPI Setting dialog. When this setting is off, non-DPI-aware applications get DPI virtualization via bitmap stretching, which results in large but fuzzy text. Turning on the checkbox disables this bitmap stretching, so non-DPI-aware apps get smaller text but without the fuzzy stretching. I find most apps to be more readable with the smaller crisper text, so I enable XP style scaling (i.e. disable DPI virtualization).  Windows 8.1 RTM has removed this option completely. After upgrading, all of my non-DPI-aware apps are running with ugly bitmap stretching. I'm able to fix this by going into the Properties for each app and setting the "Disable diplay scaling on high DPI settings" box in the Compatibility tab, but that is a bit of a pain.  Is there any way to restore the systemwide XP-style DPI scaling in Windows 8.1? I would like to make a debian system where you can only open one GUI program at boot, no other graphical interface, no minimize, or any X apart from that program, is there a way to do it on debian if possible, or any other custom distro? I just want to boot open the program and allow the user to only see and use that program. How do you add more than one material to a mesh?   I have tried adding two materials to the object, and then assigning one of the materials to a vertex-group, but I could not find a place to assign it.  I am making a striped bouncy ball that I want to deform, so adding a different material by separating the mesh into different objects would not be ideal. Here's what is happening; everytime I go to Stack Overflow (stackoverflow.com, NOT meta) with Firefox, it gives me the "Welcome back" tab that pops up from the top. Whenever I click in the "click here to refresh the page" link, all it does is refresh the page without logging me back in.  Even if I try to log in using the "log in" button at the top, all it does is pop up the tab that says "Welcome name here, you're being redirected." The problem is I'm not being logged in, not to mention that it keeps refreshing the page over and over again without giving me a choice to log in with Facebook.  For some reason, I was capable of logging in to Meta Stack Overflow using Firefox. I am still unable to log in to Stack Overflow. Any ideas? How to make math font huge How to get image size (height & width) using JavaScript? How do Prestige Classes that advance spellcasting interact with racial spellcasting ability? For example Lets use the Marrutact from Sandstorm, it can cast as a 5th level wizard. if it then takes a prestige class that requires the ability to cast 1st level Arcane spells and advances arcane casting such as Abjurant Champion. assuming it meets all other prerequisites does the Marrutacts spellcasting improve when it takes a level in Abjurant Champion or does it gain no benefit? \textcite macro produces "Author (year)" correctly but colours first paren with hyperref How are cellphone IP addresses assigned? Invisible space when trying to remove file (but exclude all files in a list)? I'm trying to accomplish a simple task using the excellent post : I would like to remove all files from a folder except those listed in a .txt file contained in the folder. The syntax works fine, but for some reason all files are removed - basically the script is saying that there are no correspondences between the filenames in the folder and those in the .txt file - but I can see that there are!   I thought maybe there might be some invisible white space after my file names that I cannot see (in the folder or the .txt file). But I don't think so...Is there another solution that I am not thinking of? I have noticed that while asking questions and scrolling to the bottom of the page, the right sidebar containing recent questions covers the footer partially. Also, a part of it goes beyond the browser's boundary. It may not be a big problem but it still doesn't look good.    This also happens on the  page