Definition: A topological space $X$ is called Hausdorff  space if for each $x_1,x_2 \in X$ (they are distinct) we can always find neighborhoods $U_1,U_2$ of $x_1,x_2$  such that $U_1 \cap U_2 = \varnothing $.  Is this definition equivalent to say that the following set   $$ \mathcal{D}  = \{ (x,x) : x \in X \}$$  is closed in $X \times X $ ??  Im having difficulties seeing why this istrue. Any help would be greatly appreciated.
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...
Compile code using JavaFX 2.0 (using command line) Why did Doc Brown choose the Delorean DMC-12 as his first time traveling vehicle? According to , the movie makers decided     The specific choice of vehicle was a  for the purposes   of it looking like an alien spaceship due to its characteristic   gullwing doors.   Is there a reason from within the storyline (comic books, cartoons, video games, deleted scenes) that explains why Emmett Brown choose that particular make and model? How to quickly and safely defrost chicken? What is the safest, quick way to defrost chicken? Microwave? Running water over it? Regular expression to match a line that doesn't contain a word I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha Grepping through .gz log files Does there exist a magical shell piping which would allow easily to grep through bunch of .gz log files without needing to extract them somewhere?  .gz files are Apache logs, result of log rotation. I'd like to quickly check how often certain URIs are accessed in the past. Is there a proof of existence of time? It seems to me that there is no such thing as time. There  is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.  Does this view of time work within the current framework of phsics?  Do physicists have an explanation/proof about time's existence? Unknown apps/processes with number only are shown in the battery screen This is a homework question. $d,n\ge 2$.  Let $L=\{(x_1,...,x_n)\in (\mathbb{R}^d)^n: x_i\in \mathbb{R}^d,  x_i\ne x_j \forall i\ne j\}$.   I tend to think it is not path connected because if you assume $d=1$, this is the case. But I cannot prove for $d\ge 2$.  Any help will be appreciated! :) I tried to use sudo cd name_of_dir but am getting the error message:   sudo: cd: command not found   Is there any other way to enter a directory owned by another user which has 700 permission? My current employer has given me the opportunity to design my own GIS workstation. The problem is I have no idea what is important when it comes to hardware! I have been given a datasheet from Fujitsu which includes the latest CELSIUS M720 series workstations but the configuration includes about 30 different processors, graphics cards, and hard drives.  I will be working a lot with the ESRI desktop suite (spatial analyst and 3D analyst) and doing a fair amount of raster processing (DEMs, visibility analysis, etc.), but I also use the GDAL library, SAGA-GIS, Quantum GIS and others.  Does ArcGIS 10 actually use multiple cores?  I have the choice from 4 cores with 8 threads up to 8 cores with 8 threads - 16 threads?  Is SSD SATA III better than HDD SATA III?  The Celsius R920 supports up to 512 GB RAM. But what do I need?  It will probably be the last time I get a new workstation for the next 5 years, so I am curious what others have ordered or feel will be necessary for the coming years. Obviously the cloud isn't taking over as quick as they say! Given a subset $A$ of $\mathbb{R}^{n}$, a point $x \in \mathbb{R}^{n}$ is said to be in the boundary of A if and only if for every open rectangle $B\subseteq\mathbb{R}^{n}$ with $x\in B$, $B$ contains both a point of $A$ and a point of $\mathbb{R}^{n}\setminus A$.  My question is from Spivak's Calculus on Manifolds: Construct a set $A \subseteq [0,1]\times [0,1]$ such that $A$ contains at most one point on each horizontal and vertical line but has boundary equal to $[0,1]\times[0,1]$. One click to site meta? Right now from a main site, it takes TWO CLICKS to get to http://meta.SITE.stackexchange.com/ - ONE!  TWO!    Why not ONE click on the &quot;Hot Meta Posts&quot; &amp; &quot;Featured On Meta&quot; - Every once in a while, someone comes to Meta to report a bug about something not working and it turns out that person is visiting one of the network sites over HTTPS/SSL. The problem has been growing since browsers started to outright block mixed content instead of showing a warning.  The usual answer is that HTTPS is not yet fully supported, and the person should try again using HTTP. Sometimes things get fixed, improving the situation a bit.  Is there a roadmap for full support? Partial support has been there for  now.  Is this actively being worked on? If not, what are the reasons to partially support it instead of disabling it completely to avoid these problems? Dynamic date range query I'm trying to create two cells that will update my query: A >= date"A1" and A &lt;= date "B1".  My goal is to create a search page for the data, where the end user could insert a date or the =today() function and it would update the query. I am currently getting a formula parse error from:  =QUERY(Data!6:1397,"SELECT A,B,C,D,E,F,G,H,I,J,K,L WHERE A &gt;= date"A1"and  A &lt;= date"B1"") What does the number in parentheses shown after Unix command names in manpages mean? The choice between the gerund and the infinitive in a certain construction I am pretty much sure that for native speakers the issue I am going to bring up might look as an uncalled question as they can easily figure out which form of a verbal part of speech should be used, be it a gerund, an infinitive, or a bare infinitive. However, it can be pretty much misleading for a foreigner because the same construction works differently with different words and I don’t see a logical explanation. For example:  The purpose of this article is to analyze the issue…  My hobby is reading. One may assume, and many foreigners really come to this conclusion, that it’s OK to switch the “gerund” and “infinitive” as the meaning will stay the same, that is:  The purpose of this article is analyzing the issue… (same as 1 above)  My hobby is to read. (same as 2 above)   However, as I was told, this assumption is erroneous. Another example: His task was watching after them. (not good) His task was to watch after them. Is there any principle which governs the choice of the right form in such cases? Standard example where Jacobson radical not equal nilradical How can I output colored text to the terminal in Python? I want to find the interval of convergence of $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(2x)^n$$ By the root test, $\sqrt[n]{|(\frac{n}{n+1})^{n^2}(2x)^n|}=|(\frac{n}{n+1})^{n}(2x)|$  And $(\frac{n}{n+1})^n=(1-\frac{1}{n+1})^n\to\frac1e$ as $n\to\infty$  So I've got an open interval(of convergence) s.t. $|\frac1e 2x|&lt;1\implies |x|&lt;\frac e2$.  Now I need to figure out whether the series is convergent or not at the end points, $x=\frac e2, -\frac e2$.  So there are two series I need to deal with, namely, $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$ and $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(-1)^ne^n$.  Suppose $(\frac{n}{n+1})^{n^2}$ is increasing. Then $$(\frac{n}{n+1})^{n^2}&lt;(\frac{n+1}{n+2})^{(n+1)^2}\iff 1&lt;(\frac{n+1}{n+2})^{(n+1)^2}(\frac{n+1}{n})^{n^2}$$ $$\iff 1&lt;(1-\frac{1}{n+2})^{2n+1}(1+\frac{1}{n^2+2n})^{n^2}$$  But $(1-\frac{1}{n+2})^{2n+1}\to\frac{1}{e^2}$ and $(1+\frac{1}{n^2+2n})^{n^2}&lt;(1+\frac{1}{n^2})^{n^2}\to e$  So I have $1&lt;\frac1e&lt;1$, which is a contradiction. Hence $(\frac{n}{n+1})^{n^2}$ is not increasing. But I don't know if it's decreasing. (is it true that $(1+\frac{1}{n^2+2n})^{n^2}\to e$?)  If $(1+\frac{1}{n^2+2n})^{n^2}\to e$ is true, then $(\frac{n}{n+1})^{n^2}e^n$ converges to some positive number, so $\sum(\frac{n}{n+1})^{n^2}e^n$ diverges. But I don't know if it's decreasing: can't apply the alternating series convergence test(Leibniz's criterion).  So, can you help me with the two end points? Do you have any useful awk and grep scripts for parsing apache logs? I can use log analyzers, but often I need to parse recent web logs to see what's happening at the moment.   I sometimes do things like to figure out top 10 ips that request a certain file  cat foo.log | grep request_to_file_foo | awk '{print $1}' |  sort -n | uniq -c | sort -rn | head   What do you have in your toolbox?