It'd be great if someone checked the proof I did for the following problem:  $f:X\longrightarrow Y$,   $f(\overline{A})\subseteq \overline{f(A)},\forall A\subseteq X \Leftrightarrow f$ continuous  proof:  Suppose $f(\overline{A})\subseteq \overline{f(A)}$ for any $A\subseteq X$ .  Let $C\subseteq Y$ be closed and define $A=f^{-1}(C)$.  $f(\overline{A})\subseteq \overline{f(A)}\subseteq\overline{C}=C$, therefore $\overline{A}\subseteq f^{-1}(C)=A$,  so $A$ is closed.  *Conversely, suppose $f$ is continuous.    Let $p\in\overline{A}$ and let $V$ be an open neighborhood of $f(p)$. Since $f$ is continuous,  $f^{-1}(V)$ is open. On the other hand $p\in f^{-1}(V)$, but since $p\in\overline{A}$ we have that $f^{-1}(V)$ contains points of  $A$. Therefore $V$ contains points of $f(A)$. Since $V$ is an arbitrary neighborhood of $f(p)$, we have $f(p)\in \overline{f(A)}$. So finally, $f(\overline{A})\subseteq\overline{f(A)}$.  Thank you very much!
As a part of self study, I am trying to prove the following statement:  Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where $\overline{A}$ denotes the closure of an arbitrary set $A$.   Assuming $f$ is continuous, the result is almost immediate. Perhaps I am missing something obvious, but I have not been able to make progress on the other direction. Could anyone give me a hint which might illuminate the problem for me?
CUDA Driver API vs. CUDA runtime How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices. Extra slash added to the chat room URLs in the side bar There is a extra slash added to the chat room URLs in the side bar. This is not causing any redirection issue or 404 not found error. Just observed it and want to notify to the development team about this. Seems it is not a bug, but that can . Currently the following URLs are binded against the chat rooms in the side bar.  Tavern on the Meta https://chat.meta.stackexchange.com//rooms/89  The Fire Department https://chat.meta.stackexchange.com//rooms/1181   GIF for the reference: How can I check which DNS server am I using (in Linux)? I am using network manager and a wired connection to my university's LAN. (I am trying to find out why my domain doesn't get resolved) Is the solution to a question regarding four smaller masses on a rotating hollow sphere accurate? I was doing the question :     A uniform 8.40-kg spherical shell, 50.0 cm in diameter has four small   2.00-kg masses attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the   center of the sphere and two of the small masses (see ). What friction torque is needed to reduce its angular speed   from 75.0 rpm to 50.0 rpm in 30.0 s?   I understand how to get $\alpha$, and how to get $\tau$ from $I$ and $\alpha$. What I do not understand is their computation of the moment of inertia. The question did not provide any information regarding the shapes of the masses, but the solution (which writes $I=\frac{2}{3}MR^2+2mR^2$) seems to assume that the masses are point particles. Thus, the two masses along the rotational axis have a moment of inertia of 0 ($r=0$), and each of the other two masses has a moment of inertia of $I=mr^2 =mR^2$, where $R$ is the radius of the sphere and $r$ is the distance from the axis.  But why are the masses considered to be point particles? I wanted to use the parallel-axis theorem to determine the moment of inertia about the rotational axis; not knowing the shape of the masses made this impossible, since I couldn’t find the moment of inertia about the CM. If I knew $I_{CM}$, then couldn’t I use $I_{axis}=I_{CM}+mR^2$ for the two masses not on the rotational axis and $I_{CM}$ for the two that are? That would make $I_{total}=\frac{2}{3}MR^2+4I_{CM,\:mass}+2mR^2$ Would that not be a better way of solving this problem? If I have NM cable running into a junction box, can I strip the sheath of the NM cable starting inside the junction box and run the conductors without a sheath through conduit that is connected to the junction box? I understand that the wiring you'd typically use through conduit is THHN or NM (if you want to use the conduit solely as protection).  But I'd like to be able to take this incoming NM cable and have it's conductors only continue on through conduit along with some other THHN that's coming from another source. The individual conductors of the NM cable while not labeled can still be traced back to where the sheathing ends, so technically the conductors are still identifiable as to what they are. If need be, I could splice the NM conductors to THHN wire in the junction box, but I don't see a reason to add a splice where I don't think it's needed. My reasoning for why I think this should be OK is that if you terminate NM cable in a panel, you typically have a foot or so of the conductors without the sheath and there's nothing identifying those other than tracing it back to the sheath it came from. It would be nice to see what users have recently been active. This is somewhat standard with many forum-style websites, showing how active the site has been in the last n-minutes or so. As a functional advantage, this also serves to let you know when is a good time to get a question in. A word that means stage-fright, but for a person? Suppose you met a famous person, and you were shy and anxious and almost overwhelmed by them.   Is there a word for describing that? Similar to getting stage fright, or "choking", except in this case for a person?     When he said hello to me, I couldn't respond. I (was) __________. Is there a way to configure pdflatex to make all but the .tex and .pdf files hidden files? I have asked a  about how to manage all the files generated when my LaTeX files are compiled but the approach in the answer I accepted is still unsatisfactory for the many small documents I generate.    I realize the LaTeX  all the files it generates so I can't prevent their creation.    However is there a way to configure pdflatex to make all but the .tex and .pdf files hidden files (i.e., preceded by .)?   (Edit: The two main systems that I use are tex live on Ubuntu Linux and BasicTeX on OS X) How to use EntityFieldQuery with dates? I need to select entities with specific dates.   How should the following be formatted. Do I need to parse 15-May-2010  Also how I get the error page.   function events2() {  $query = new EntityFieldQuery(); $query   -&gt;entityCondition('entity_type', 'node', '=')   -&gt;propertyCondition('status', 1, '=')   -&gt;propertyCondition('type', 'event')     -&gt;propertyCondition('field_event_date', '15-May-2010', '=');  $result = $query-&gt;execute();   return $result;  } Transfer from JFK Terminal 2 to Terminal 4 I have family who does not speak English.  They are changing planes in JFK Airport and will most likely be transferring from Terminal 2 (Delta) to Terminal 4 (China Southern Airlines).  At first I thought that directing them to use the Airtrain would be best.  However, in doing more reading my understanding is that the Airtrain would mean they would need to go through security again.  I found this on Delta's site:   If they use the JITNEY Shuttle to change from Terminal 2 to Terminal 4, does that mean they would not need to re-clear Security?  Edit: They are flying TO China. They are going from Tampa -> NY JFK -> China. What is this Windows 8 icon for as rectangle icon in the title bar? Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that:  $$f(x)f \left(\frac{1}{x}\right)=1$$ In our campaign we have several characters who never remove their armor. They wear it while adventuring, sleeping, swimming, bathing, etc.  Are there any penalties for:   Wearing armor while sleeping? Wearing armor while swimming?   If there are penalties, do they change based on the category of armor (light, medium, heavy)?  I am primarily looking for RAW penalties specified for armor, but I'm also open to applications of other rules (eg - higher DC for swimming in heavy armor, disadvantage during some situations). If I short sell stocks of a company but do not close my position by buying them back, what will happen? I realize it is a bad thing, but what exactly? I will need to pay a penalty? My current understanding:  $\vdash$ is "syntactic consequence", e.g. if we have $a \vdash b$ it means if we've written down $a$ as something we know, we can immediately write down $b$ as something we know. This is saying nothing about whether any of these pieces are true, false, something else, etc, it's just saying that we can say something about $b$ by virtue of being able to say something about $a$.  $\vDash$ is "semantic consequence", e.g. if we have $a \vDash b$, then it (usually) means if $a$ is always true in every interpretation (i.e. no matter how we mess with any of the underlying variables driving $a$'s valuation), then $b$ is true as well. However I don't know if this is necessarily just a "true"-only thing. It seems conceivable that we could make one for falsehoods too, i.e. if $a \vDash b$ might mean if $a$ is always false then $b$ is always false, of if $a$ is always "dog" then $b$ is always "dog", etc. But I am unsure of this one.  $\to$ is material implication, which exists solely within the logic system (as opposed to the previous two metalogical relations) and has a specific semantic meaning / definition in terms of a truth table in something like prepositional logic ($a \to b$ is always true unless $a$ is true and $b$ is false). It's defined this way simply because it is a helpful / useful model to describe things like "whenever $a$ is true, $b$ is true".  Do I have the right idea so far? How to decide if an effect is magical for Magic Resistance? Some creatures have Magic Resistance:     The [...] has advantage on saving throws against spells and other magical effects.   We know from  that this works against Turn Undead, but does it work against Stunning Strike, Death Strike, Fey Presence, etc? How do we decide? How to solve "sudo: /etc/sudoers.d is world writable" I'm trying to make a basic Windows application that builds a string out of user input and then adds it to the clipboard. How do I copy a string to the clipboard using Python? How to open and convert CHM documents? I have some documents that are in a .chm format. I wondered if there is a file format that can be easier to navigate, supported and of equal file size in Ubuntu?  If there is, I would like to start converting all those books and probably using them with less hassle on all my Ubuntu PCs and my Android phone.