Is a dense subspace of a separable topological space also separable?
Is every dense subspace of a separable space separable?
What do you call the process of combining two words to create a new one? Why do we need both using namespace and include directives in C++ programs?  For example,   #include &lt;iostream&gt;  using namespace std;  int main() {  cout &lt;&lt; "Hello world"; }   Why is it not enough to just have #include &lt;iostream&gt; or just have using namespace std and get rid of the other?  (I am thinking of an analogy with Java, where import java.net.* will import everything from java.net, you don't need to do anything else.) Every time a user posts something containing &lt; or &gt; in a page in my web application, I get this exception thrown.  I don't want to go into the discussion about the smartness of throwing an exception or crashing an entire web application because somebody entered a character in a text box, but I am looking for an elegant way to handle this.  Trapping the exception and showing     An error has occurred please go back and re-type your entire form again, but this time please do not use &lt;   doesn't seem professional enough to me.  Disabling post validation (validateRequest="false") will definitely avoid this error, but it will leave the page vulnerable to a number of attacks.  Ideally: When a post back occurs containing HTML restricted characters, that posted value in the Form collection will be automatically HTML encoded. So the .Text property of my text-box will be something &amp; lt; html &amp; gt;  Is there a way I can do this from a handler? How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$ How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$?  I have already recognized one problem caused from this: it  follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct a function $f(x)$?  Best regards and thanks, John How do I replace this cast iron toilet flange? I have a material assigned to a few test objects with an image sequence as a texture. If I view the scene in render preview, and scrub the timeline, (or render a sequence,) the texture behaves as expected.  However, with the viewport set to 'texture' or 'material' I only see the first frame of the sequence, wherever I am on the timeline. 'Auto Refresh' is set in the cycles texture node. Is this the expected behaviour? find $\sum_{k=2}^{\infty}\frac{1}{k^2-1}$ How to detect, from browser, if user is running in Remote Desktop session? How to access object attribute given string corresponding to name of that attribute What is a simple example of a limit in the real world? This morning, I read Wikipedia's informal definition of a limit:     Informally, a function f assigns an output $f(x)$ to every input $x$. The   function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever   $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$   as $x$ moves closer and closer to $p$.   To me that sounds like something that might be better described as a 'target'.  If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).   So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.   What is a simple example of a limit in the real world?  Thank you  -Hal. The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding to this singular value?   Usually I found the columns of U with this relation, so let's say that we refer to $\vec{u_2}$ as the U column vector corresponding to singular value equal to 0:  $\vec{u_2} = (1/s_2)Av_2$  Now the as we can see the 0 is in an illegal place, since you can't divide by zero. But if you move it to the left side, I still can't quite see how to find the column of U corresponding to it. Any suggestions?  Here is an example of a matrix that provided this specific challenge  $ A = \left(\begin{array}{rrr} -3 &amp; 1 \\ 6 &amp; -2 \\ 6 &amp; -2 \end{array}\right). $  $ \lambda{_1} = 90$,  $\lambda{_2} = 0$,  $s_1 = 3 \sqrt{10}$,  $s_2 = 0$,  How can one find $\vec{v_2}$ when $\lambda{_2} = 0$? In this case, you know that the eigenvectors of $A^{t} A$ are orthogonal, solve for $\lambda{_2}$ and find the eigenvector if it's orthogonal to the eigenvector corresponding to the first eigenvalue you have your solution.  So we end up with the following normalized column vectors for V, which we can see are orthogonal:  $ \vec{v_1} = \left(\begin{array}{rrr} -3/\sqrt{10} \\ 1/\sqrt{10} \\ \end{array}\right). $  $ \vec{v_2} = \left(\begin{array}{rrr} 1/\sqrt{10} \\ 3/\sqrt{10} \\ \end{array}\right). $ How to suppress Terminated message after killing in bash? How can you suppress the Terminated message that comes up after you kill a process in a bash script?  I tried set +bm, but that doesn't work.  I know another solution involves calling exec 2&gt; /dev/null, but is that reliable? How do I reset it back so that I can continue to see stderr? Motion tracked Video doesn't show up in the render or in the compositing tab I tried to add a 3D object to a clip by following a youtube video. But clip doesn't show up in the render or in the compositing tab. It show up even in the camera view. I didn't change any settings. please help. Thank you.    And the  rendered images has only the cube in it like in the picture above. 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. Should edits with only minor code improvements be accepted. As an example, I'm reviewing an edit that changes all double quotes to single quotes and renames a variable in some javascript. With the exception of the variable rename (which is not entirely necessary), the edit is frivolous. Should I accept on the merits of a debatable variable name change? Line numbering paragraph oriented? I'm using the lineno package with the pagewise option but I need a way to get the numbering restart every paragraph break. I need a general solution to pass to the latex command, e.g.:  latex "\AtBeginDocument{\RequirePackage[pagewise]{lineno}\linenumbers}\input{mydocument.tex}"   I read the lineno manual but the package does not provide this feature. Is there any trick to do that? How do I check if an integer is even or odd? Changing Login Logo Picture/Video capture programs What software options are there for taking videos and or pictures within Ubuntu (Much like Apple's Photobooth)?  Please limit one software solution per answer and include key features/description of the software How to tweet using terminal?