I can sort of see this intuitively, seeing as it's a similar argument to the Baire Category theorem, but does anyone have a proof I could look at?   Would it suffice to say that every locally compact Hausdorff space is also a Baire space $\implies$ not the countable union of nowhere dense sets?  If so, how about this for a proof?     Since X is locally compact Hausdorff, it is    homeomorphic to an open subspace of a compact   Hausdorff space, say Y. By the Baire Category    Theorem, Y being a compact Hausdorff space   implies that Y is a Baire space. Since    open subspaces of Baire spaces are Baire spaces   Then X is a Baire Space.   This is under the assumption that a Baire space $\implies$ not the countable union of nowhere dense sets? Unless I misinterpreted the Baire Category Theorem.
I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.  The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.  How can I use this on the proof?
Headers and font in KOMA-script I am setting up the layout for my bachelor thesis using scrbook as basis for a self-made documentclass with twosided print. I want a header containing the current part number and name on the outer side (left on even pages, right on odd pages), the current chapter number and name on the center of the head and the same for section on the inner side. I found this example in the KOMA-Script documentation:  \usepackage{scrlayer-scrpage} \pagestyle{scrheadings} \automark[chapter]{chapter} \automark*[section]{}   It puts the chapter on the outer side on even pages and the section on the outer side on odd pages. Sadly, I was not able to modify this to fit my needs because I do not really understand the way KOMA-Script works here. If a part starts but there is still no chapter in it, the place for the chapter shall remain empty (same for chapter and section). Can anyone help?  Furthermore, I would like to change all fonts that KOMA-Script changes (e.g. toc, section, chapter...) back to the default computer modern as I do not like the font KOMA-Script uses here. Is there a better way than changing all the fonts manually like this:  \addtokomafont{section}{\fontfamily{cmr}\selectfont} I want a green for "correct answer" and a red for "wrong answer". My current choice is rgb(144, 238, 144) for the green and rgb(250, 128, 114) for the red. (See wikilearner.net/wiki for the colours in action.)  I have been told that these colors are very close in contrast, making it difficult for a person with red-green color-blindness. What would be a better choice? Some GUI apps use OS X location services, but I want to retrieve a Mac's physical location from the . This could be useful for running scripts, switching settings etc. The mechanism should be , not a Geo IP service (data too low-res, needs online connection).  Minimum requirement is output of longitude and latitude. Nice would be a number indicating the precision. Super nice: a geocoder that provides location names. Avoiding Mixed DML (setup/non-setup) Approach I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... What is the maximum number of downvotes a question can have I was just wondering what the maximum number of down votes a question can have before it is removed from StackOverflow. Which is correct: "one or more is" or "one or more are"? How can you encode a string to Base64 in JavaScript? I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. We are using Nginx to serve static files on a development platform. As it is a development platform, we'd like to disable caching so that each change is propagated to the server. The configuration of the VHost is quite simple:  server {   server_name  static.server.local;   root /var/www/static;    ## Default location   location / {     access_log        off;     expires           0;     add_header        Cache-Control private;   }  }   When we access an HTML file (http://static.server.local/test.html), we have no issue: the server returns a code 304 Not Modified as long as the file is not changed, and a 200 OK response with the modified file when the file is changed. However, it seems to behave differently with a Javascript or a CSS file. Once the file is changed, we get a 200 OK response as expected, but with the old text. Is there an internal cache mechanism in Nginx that could explain this behaviour? Or some configuration that we should add?    As a side note, here is the header returned by Nginx when the file has been modified (it seems correct):  Accept-Ranges:bytes Cache-Control:max-age=0 private Connection:keep-alive Content-Length:309 Content-Type:text/css Date:Fri, 13 May 2011 14:13:13 GMT Expires:Fri, 13 May 2011 14:13:13 GMT Last-Modified:Fri, 13 May 2011 14:13:05 GMT Server:nginx/0.8.54   Edit After trying different settings with the expires directive and Cache-Controlheader, I made some further investigations. In fact, the server is installed on a VirtualBox guest Ubuntu, and data are read from a shared folder that is on the Mac OSX host. If the file is edited from an IDE (NetBeans) on the host, it seems that changes do not appear whereas if I edit it directly on the guest (using VIM), it is refreshed. The strange thing is it does not behave similarly with HTML files. Quite puzzling.  Edit 2 (ANSWER) Indeed, the origin of the issue was more on the VirtualBox side. Or rather a conflict between VirtualBox and the "sendfile" option of the server. This link  gave me the solution: switch the sendfile flag in the server configuration to off:  sendfile  off;   Hope this could also help other person using VirtualBox for development. :) There are some additional information on the . How to restart the networking service? I tried to use  sudo service networking restart   and  sudo /etc/init.d/network restart   but they both crash the window manager and I can no longer use my keyboard for input into X.  when I use the /etc/init.d/ method it complains saying that I should use the service utility  e.g. service networking restart   but it crashes just the same.  Is there a GUI method of restarting networking? What is the difference between correlation, causation and prediction? Suppose we have a set of events $\Omega$, containing events $A$ and $B$. My econometrics professor tried to distinguish the following three terms today.    Causation --- $A$ causes $B$ if the occurrence of $A$ always leads to another specific outcome $B$. For example, clapping my hands causes a sound to be emitted.  Prediction --- $A$ predicts $B$ if on average, $B$ is the expected outcome from $A$ occurring. In other words, whereas causality is deterministic, prediction is probabilistic. For example, studying for a test would predict doing well on the exam. But there's no guarantee.   Correlation --- $A$ and $B$ are correlated if when one occurs the other does too. But there may be some hidden variable causing both.    Are these distinctions meaningful? Or did he make these terms up?   My Questions   What is the difference between causation and correlation?  Are "causes" always deterministic? Ideal contained in a finite union of prime ideals Let $I \subset R$ be an ideal and $P_i$  $(i=\{1,...,n\})$ prime ideals with $I\subseteq\bigcup_{i=1}^nP_i$. Prove that then $I$ is contained in one $P_i$.   I don't know how to show this because I don't have any approach.  So I am looking for something to start with or something like a sequence of tips (Since I know this is a large proof)  Thanks in advance! Every Finite Group is Isomorphic to a Subgroup of $A_n$ What is the 'rev' dropdown supposed to do? I only see this on sites where I don't have edit privileges, and only where there have been multiple revisions:        What exactly is this supposed to do?  Intuitively, I would expect it to change the text in the box to what it was at that revision, but it doesn't. And, if that's it, why only where I need to suggest edits? $X$ is a topological space with no special properties.  If a space $X$ has no isolated points, then show the same for any dense subset of   $X$.  Thanks a lot. Why does ice melting not change the water level in a container? I have read the explanation for this in several textbooks, but I am struggling to understand it via Archimedes' principle. If someone can clarify with a diagram or something so I can understand or a clear equation explanation that would be great. Imagine that a pebble is placed on a uniformly rotating, frictionless disk.  What will happen to this pebble? Will the disk slide under it and the pebble stay as is? Or will there be a centrifugal force on the pebble and it'll be thrown off the disk? A does the same B as does C Export nodes from Drupal 4? I have auto traced a photograph in illustrator - its got thousands of vectors once I expand it and illustrator has converted each of them to a filled colour - I want to now trace the edges of them with a grungy brush but using the underlying fill colour. Where in illustrator can I stroke or outline these fills using the underlying colour - surely this is something that the program automatically does?   I've found this script online but don't know how to implement it. I'm new to this so any help appreciated. Many thanks.   James   var idoc = app.activeDocument;   var sel = idoc.selection;    for (i=0; i&lt;sel.length; i++) {       var ipath = sel[i];       ipath.stroked = true;       ipath.strokeColor = ipath.fillColor;   } Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers. Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers.   This is question 5 from Herstein Page 35. I have proved that conclusion holds for three positive integers.Thanks.