Let X be a topological space. Show that X is $ T_2 $ if, and only if, $ \alpha $ = {(x, x) $ \in $ X $\times$ X : x $ \in $ X} is closed at ... Let X be a topological space. Show that X is $ T_2 $ if, and only if, $ \alpha $ = {(x, x) $ \in $  X $\times$ X : x $ \in $ X} is closed at  X $\times$ X. =================================================================== By Hausdorff definition we have x, y $ \in $ X, with x $ \neq $ y, there are U, V $ \subset $ X open such that x $ \in $ U, y $ \in $ V e U $ \cap $ V = $ \emptyset $. Assuming that close in X $ \times $ X = A and that to be close $ A^c $ needs to be open. But if X is Hausdorff that means that exist x $\in$ X, U $\subset$ X open, where x $\in$ U. But that doesn't mean X $\times$ X is open?
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it...
Evaluate definite interval of normal distribution Getting Windows 7 admin account password? Can you teleport farther away with misty step if you have a double from Invoke Duplicity? The Channel Divinity feature says:     you can cast spells as though you were in the illusion's space, but   you must use your own senses. (PHB 63)   If you were where your duplicate is, you could step to a space 30ft from its position. So if it is 60ft from you, you could teleport to a space 90ft from your actual position. But the spell says that:     you teleport up to 30 feet (PHB 260)   This would put a hard limit on the distance you can travel.  Which rule takes precedence here? This is something that I always find a bit hard to explain to others:  Why do XML namespaces exist?  When should we use them and when should we not? What are the common pitfalls when working with namespaces in XML?   Also, how do they relate to XML schemas? Should XSD schemas always be associated with a namespace? It would be nice to allow users to mark a change in an answer (or question) as minor (e.g. for a simple spelling or grammar correction). A change marked as minor wouldn't push the question on top of the list of questions on the home page or in the feeds. It wouldn't trigger any notification. This could be implemented with a "Minor change" checkbox when editing an entry (a la Confluence). What do you think? Has a free neutron ever been shown to absorb/emit/interact with a photon? Protons only 'interact with' very high-energy photons, whether inside a nucleus or free, right?  I'm assuming the same about neutrons....  Neutrons have a small magnetic moment and a slight electric 'moment' (dipole?), correct?  They are sometimes considered to consist of a negative charge surrounding a positive charge?  I would assume that neutrons, in a nucleus or free, interact with photons even less often than protons, and that the gamma photon has to have an even higher energy than one interacting with a proton....  Lastly, I have read about 'photodisintegration', in which a nucleon is kicked out of a nucleus by a gamma ray.  It seems to happen to neutrons sometimes...  But, I am mostly curious about experimental evidence of free neutrons interacting with photons, how often it happens compared to free protons, and whether the interacting photon's energy has to be extremely high, higher even than a photon interacting with a proton... With the new  and the upcoming release of  film, JK Rowling has been awesome and given us new writing on the .  In the introduction it is stated that there are eleven established/registered schools. There are writings for six of them. I don't think we need a new writing for Hogwarts... so that means seven have been located and defined.   Hogwarts School of Witchcraft and Wizardry - Great Britain  - France  - Northern Europe (possibly Estonia, Latvia, or Belarus)  - Japan  - Africa (possibly Congo/DRC)  - Brazil  - United States   Do we know anything of possible location or names of the other four schools? I know Koldovstoretz was mentioned as school in Russia, but is that still canon?   Koldovstoretz - Russia (maybe depending on current canon)   The final three are my guesses based on the population and geography on the world.   Name Unknown - China (based on population) Name Unknown - India (based on population) Name Unknown - Australia (based on geography)   Related: Why does HMAC use two different keys? Suppose $H$ is a hash function; why is $$H(k\mathbin\|H(k\mathbin\|m))$$ not secure?  See this . In there, indeed two keys are used and the mac algorithm is $$H(k_1\mathbin\|H(k_2\mathbin\|m)).$$ Why don't we use $$H(k\mathbin\|H(k\mathbin\|m)),$$ which has only one key? How to parse hundred html source code files in shell? I came across a  where a comprehensive example would be very useful. So I wrote an example program showing one way to solve their problem.  My problem is that when I was done creating it, and testing to make sure it worked, it was over 400 lines in 18 files. My example included source code, header files, and scripts for building the example in a variety of circumstances.   It's a working and tested, so it's of huge value to illustrate the tricky solution, but because it's all there, my response was huge! Is there a better way of posting my code dump?  Edit:  I guess there isn't a current "best way" to include a large chunk of code, so I made this a feature request and tried to add some ideas.  Perhaps code entries that are longer then a certain number of lines, or ones that have been specifically marked could initially be displayed as "collapsed", and someone could expand them if they want to see them?  Or maybe, like attaching images, users with enough reputation could attach an archive of files. They could be displayed as "attached code" (initially collapsed also).  Some problems have simple answers that unfortunately touch disproportionately large parts of the program, or can be solved in slightly different ways depending on the environment. I feel that stackoverflow needs a way of capturing these large solutions. Sometimes, they're large chunks of code. In my case, the code was small, but there were many files.  Using github or a similar code repository is great, but we risk linkrot (even for big guys like github) and loose the community editing, commenting, and voting abilities.  I understand the problems that could come from with people dumping whole "I can't get this to work" packages, but surely we already have a solution for that in moderation.  Edit again:  I'm starting to see that this may have been already addressed in posts like    , and    If you feel this a dupe, I'm open to having it "closed as already addressed" Down vote changed to up vote caused reputation meter to pass daily maximum Checking if file is checked out through CSOM I'm a little confused about the output I'm seeing in my routing table, particularly the 'metric' column:  $ route -n Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 0.0.0.0         192.168.0.1     0.0.0.0         UG    0      0        0 wlan0 172.16.35.0     0.0.0.0         255.255.255.0   U     0      0        0 vmnet1 192.168.0.0     0.0.0.0         255.255.255.0   U     9      0        0 wlan0 192.168.82.0    0.0.0.0         255.255.255.0   U     0      0        0 vmnet8   According to the man page, metric indicates the 'distance' to the target. I was a little confused about what exactly "target" was. I assumed it was my router (it goes to gateway 0.0.0.0, which then goes to my router at 192.168.0.1). Thus, I expected the metric to be a single hop to my router. However, it's 9! Why is this so number high? How can author deal with late reviews? Another problem I personally struggle with in peer-review publication process is the situation when review takes really, really, really long. The precise definition of "long" is probably very area-specific: for example a paper in Maths will probably took longer to review than one in Physics. Many well-established journals and major publishers, sadly, prefer not to specify the "rules of the game" precisely, particularly how long the review process should normally take. Fortunately, I quite often serve as a referee with the very same journals and know the amount of time I am given to write a review. Unfortunately, when I am an author, I do not observe these deadlines to be always met. For example, a notable U.S. publisher would ask a reviewer to send a review in 2 months; but in practice a first review of my paper once took as long as 8 months.  As a rule of thumb, I usually start writing to editors with questions about "the status of my paper" in X+1 months time, when X is a deadline time for a reviewer to submit a review. This strategy is not particularly successful &mdash; what I usually get is a recommendation to be more patient. Since there is no "official" deadline time announced, there is no formal ground for complaints here.  Of course, such delays lead to a whole bunch of funny situations, particularly if a preprint is put online (e.g., on arXiv) and the results are presented in some conferences. Follow-ups with no official paper to cite, grant applications in which you can not justify by a solid publication, good research but bad bibliometrics or CV, to name just a few.   Could you recommend an efficient way to deal with this problem? How can I enable Ctrl+Alt+Backspace? Ctrl+Alt+← can usually be set to restart X, however I can't find the option to allow this (it is disabled by default).  I have looked in System-&gt;Preferences-&gt;Keyboard and System-&gt;Preferences-&gt;Keyboard Shortcuts but I can't find an option to enable it.  Where can I change this setting? Reverse (parse the output) of Arrays.toString(int[]) Reprojecting MODIS-Sphere ESRI GRID file to WGS84 using QGIS/GDAL I have an ESRI GRID file projected in MODIS-Sphere (here's the dataset ) that I need to convert to a WGS84 GeoTIFF.  I tried Raster > Projections > Warp (Reproject) and got this nonsense:   I then thought I'd use the MODIS reprojection tool or HEG to convert it, but they only accept HDF files and using QGIS' warp or translate functions to convert this file to HDF throwns an error. This guy was trying to do the same thing in Arc, no joy:   I looked for clues in the metadata, and the file seems a little goofy; gdalinfo yields:  Driver: AIG/Arc/Info Binary Grid Files: seto_uf        seto_uf.aux        seto_uf/.aux.xml        seto_uf/.DS_Store        seto_uf/dblbnd.adf        seto_uf/hdr.adf        seto_uf/log        seto_uf/metadata.xml        seto_uf/sta.adf        seto_uf/vat.adf        seto_uf/w001001.adf        seto_uf/w001001.tif        seto_uf/w001001.tif.aux.xml        seto_uf/w001001x.adf Size is 7468, 2911 Coordinate System is `' Origin = (-17345341.010356999933720,8423784.764957003295422) Pixel Size = (5000.000000000000000,-5000.000000000000000) Corner Coordinates: Upper Left  (-17345341.010, 8423784.765)  Lower Left  (-17345341.010,-6131215.235)  Upper Right (19994658.990, 8423784.765)  Lower Right (19994658.990,-6131215.235)  Center      ( 1324658.990, 1146284.765)  Band 1 Block=256x4 Type=Byte, ColorInterp=Undefined   Description = Seto_UF   Min=0.000 Max=101.000    NoData Value=255   So it can't determine the coordinate system.  There's a .tif included in the directory, but gdalinfo on that shows:  Driver: GTiff/GeoTIFF Files: w001001.tif        w001001.tif.aux.xml Size is 7468, 2911 Coordinate System is: GEOGCS["Unspecified datum based upon the GRS 1980 Authalic Sphere",     DATUM["unknown",         SPHEROID["unnamed",6371007,1]],     PRIMEM["Greenwich",0],     UNIT["degree",0.0174532925199433],     AUTHORITY["EPSG","4047"]] Origin = (-17345341.010356999933720,8423784.764957003295422) Pixel Size = (5000.000000000000000,-5000.000000000000000) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=BAND Corner Coordinates: Upper Left  (-17345341.010, 8423784.765)  Lower Left  (-17345341.010,-6131215.235)  Upper Right (19994658.990, 8423784.765)  Lower Right (19994658.990,-6131215.235)  Center      ( 1324658.990, 1146284.765)    So it says the units are degrees but those huge corner coordinates and origin values don't agree.  Any ideas?  My setup: OSX 10.9.4 QGIS 2.6.1 Brighton GDAL 1.11.2 (apparently there are newer versions but this is the latest you can get on OSX without having to compile it yourself) How to loop the particle simulation indefinitely? How to determine the point of intersection of two lines? I have two lines say P1( 0, -1, 0, -1 ) and P2( -1, 0, 0, -1 ). Since I'm working in 2D, there is no Z component. The x,y,z components are normals and the w component is the distance from origin. I am not given the origin or any points on the line.  What method should I use to programmatically determine the point of intersection for these two lines? How do I get my Broadcom BCM4313 working correctly?