Is preimage of closure equal to closure of preimage under continuous topological maps?
continuous function: preimage of closure of a set is closure of preimage
Exam with 18pt main font While my code is compiling, I usually like to go through some reviewing and especially the late answers.  What I do see a lot in answers to old questions is the self-defense mechanism     ""   As if there is something fundamentally wrong with answering old questions. This leads me to two questions/thoughts:   Should we edit phrases like that out of the answer? (Given it's a good/valid answer to begin with, without any of the likely "old answer" problems. I realize the example linked to has its flaws.) Should we perhaps explicitly encourage answering old unanswered questions somewhere in the FAQ? (I'm unaware of any such recommendation, but might have read over it. Let me know if I did) A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition A set $A \subset \ell_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon &gt;0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| &lt; \epsilon$ for all $n&gt; n_0$ and $x \in A$.   It's easy to prove the part assuming that $A \subset l_1$ is compact, but I am finding difficulty to prove the other part. apt: how to be sure that a specific package gets installed/updated from specific repository and not from the default one? Is it possible to know the (public) IP address of a computer where TeamViewer is ready for a remote control session - without connecting to the computer? 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit I'm looking for a 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit. I think they were going through different dimensions. I remember part of it, but I don't think I ever saw how it ended. I remember my friend telling me there was a twist at the end. I think they were trying to find something and he was always ahead of them.. Given a joint density function, what is the conditional expectation $E[Y|x]$? The random variables $X$ and $Y$ have the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0,  \mbox{   otherwise}\end{matrix}\right.$$ Evaluate the conditional expectation $E[Y|x]$.    I'm not sure if I'm doing this correctly. Please let me know if something is off in my solution. Thanks for reading.  My solution: (My expected value is negative, which I'm pretty sure can never be the case. Also in line 4 and 8, are the domains correct?)  $$E[Y|x] = \int_{-\infty}^{\infty}yf_{Y|X}(y|x)dy.$$  $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_{X}(x)}.$$    $$f_{X}(x) = \int_{-\infty}^{\infty}f_{X,Y}(x,y)dy$$   $$= \int_{0}^{\infty}e^{-y}dy $$  $$=\left [ -e^{-y} \right]\Big|_0^\infty $$  $$ = -e^{-\infty}-(-e^{0})=1. $$ $$f_{X}(x) = 1, x \geq 0.$$ Thus,   $$E[Y|x] = \int_{0}^{\infty}-ye^{-y}dy$$  $$= -\int_{0}^{\infty}ye^{-y}dy$$ After integration by parts,  $$E[Y|x] = \left [ ye^{-y}+e^{-y} \right]\Big|_0^\infty$$ $$=0-[e^{0} + 0]=-1$$ Therefore, $E[Y|x] = -1.$  Thanks in advance for reading this long problem and responding! I find that at the end of a long day, I sometimes have a large number of programs running. All which I will need to use tomorrow. Normally, this isn't an issue, I can simply lock the machine and come back tomorrow. My problem arrises when windows update launches in the middle of the night and force restarts my computer. That in turns closes all my open software. I of course save everything regularly so I don't loose anything, but I waste time reopening all of those resources whenever there is a restart.   [EDIT]  I should clarify that I still want to be able to restart my computer when an update comes down. Preventing the restart only delays the problem until later. I should have been more specific in that I want to be able to recover my working environment after a restart for any reason. Things like scheduled maintence, power loss, updates, and software installs.  [EDIT]  I can't simply have them setup to launch at startup becasuse those files change from week to week. So I need something that monitors what I have open, and gives me the option to "recover" those software sessions when I log back in.  Anyone have any suggestions on what I can do? I'd even be willing to purchase software to do this for me if that is the only option.  Thanks I get the following error when I try to typeset a beamer document.  kpathsea: Running mktexpk --mfmode / --bdpi 600 --mag 0+525/600 --dpi 525 mathkerncmssi8 gsftopk: fatal: map file `cid-base.map' not found. mktexpk: don't know how to create bitmap font for mathkerncmssi8. mktexpk: perhaps mathkerncmssi8 is missing from the map file. kpathsea: Appending font creation commands to missfont.log.  ) !pdfTeX error: pdflatex (file mathkerncmssi8): Font mathkerncmssi8 at 525 not found  ==&gt; Fatal error occurred, no output PDF file produced!   But if I remove sansmathaccent.sty from the TeX Live tree, the document typesets fine. Here is a minimal working example.  \documentclass[10pt]{beamer}  \begin{document}  \begin{frame}{A very short beamer document}  $e^{i\pi}+1=0$  \end{frame}  \end{document} What autofocus zone selection mode should I use for action/event shooting? I have a Canon 7D, and typically shoot everything I do in its single-point autofocus zone select mode, because I usually want fine control over exactly what the camera is focusing on and I often find that its two automatic focus zone select modes don't focus on what I want in focus.  For most of the photography I do, this is fine -- I have enough time that I can compose the shot the way I want, use the joystick on the back to select the right focus point, and take the shot at my leisure.  However, recently I've been doing more event/action photography, and I've been running into issues with this mode of operation.  Too often, I will see a person start to do something interesting, but my autofocus point is not in the right place -- so by the time I get the camera up, adjust the settings, and move the autofocus point, the opportunity to take the shot has passed.  My question is, what autofocus zone selection mode and workflow should I use for this type of photography?  I could use the 5-zone partial-auto mode (which gives me a choice of top, middle, bottom, left, or right, and then the camera chooses the points within my selected region), but I find that the time needed to switch in between that mode and the single-point mode is similarly prohibitive, and I have to hope that the camera chooses the right thing to focus on. so someone tells me i need to call a few people.  they send me a vCard called contacts.vcf.  i import it.  i open my phone app and start to dial, but wait...i don't know their names.  How do i look up a recent vCard on my phone?  How do i know who's contact i've been sent?  when i navigate to a .vcf file, i can't open it.  all i can do is import it.  polaris office won't import/open it.  is there an android txt editor i can use to peruse a vCard?  yours truly, Confused by vCards Note 3, kit kat Moving linux install to a new computer I know that it can, in some circumstances, be difficult to move a Windows installation from one computer to another (physically move the hard drive), but how does that work on linux? Aren't most of the driver modules loaded at bootup? So theoretically would it be that much of a hassle?  Obviously xorg configs would change and proprietary ATI drivers and such would have to be recompiled (maybe?). Is there more to it than I'm thinking of?  Assume the 2 computers are from the same era, ie both i7s but slightly different hardware.  Update: Thanks for the answers. This is mostly for my own curiosity. I have my linux system up and running at work, but eventually I'd like to move to a computer that I can get dual video cards into so I can run more than 2 monitors. But not any time soon Show that conguence $\bmod m\,$ is an equivalence relation How can I convert a node made material into an image texture? I have a sphere having a planet material MADE WITH NODES, and I want to convert this material into an image texture; because this material is made with nodes and only works in Blender, but I want to use in another software. This is my sphere material :   For Example : This is Jupiter :  And this is the texture :  In this case I used a downloaded image for Jupiter to be the material but I want the opposite, I want to change the material to image texture Like The shape of Jupiter texture. Is there anyway in Blender that can achieve my goal ? This is the file with my sphere material .  Textures Needed :  Albedo / Color / Diffuse Normal Map Roughness Glossines  I am using normal nodes and I am not useing the new Principled-shader Is there any utility which will read a photo's date and time from its EXIF data and embed it into the corner of the photo? How does QGIS Zonal Statistics handle partially overlapping pixels? I've created a map system for a game that runs on the principle of drawing the picture of the map from tiles. There are many reasons for this which I won't go into here but if you really want to know then I'm sure you can find out how to contact me ;)  I have  so you can see exactly where the problem lies and the source. The issue is the line between the top 2 tiles and the bottom 2 tiles, I can't figure out why it's gone like this and any help would be appreciated.  In the source is a marker called "stackoverflow", if you search for "stackoverflow" when viewing source then it should take you to the table in question.  I have also uploaded an . How can I do the following in Git?  My current branch is branch1 and I have made some local changes. However, I now realize that I actually meant to be applying these changes to branch2. Is there a way to apply/merge these changes so that they become local changes on branch2 without committing them on branch1? Removing spaces between "words" in math mode Get OS-level system information