Let $p:\overline{X}\rightarrow X$ is a continuous mapping. A continuous map $s:X\rightarrow \overline{X}$ such that $p\circ s =Id_X$ is called a section of $p$.  Suppose $\overline{X}$ is connected also. Suppose that $p$ is a covering projection. Show that any section of $p$ is a homeomorphism onto $\overline{X}$.  As $p\circ s=Id$ we see that $s$ is injective. Suppose we can prove that $s$ is surjective.    Then we have a bijective continuous function $s$ such that $s^{-1}=p$ which means that $s$ is homeomorphism. $$p\circ s =Id\Rightarrow p\circ s\circ s^{-1}=s^{-1}\Rightarrow p=s^{-1}.$$  So, enough to prove $s$ is surjective i.e., $s(X)=\overline{X}$.  As $\overline{X}$ is connected, showing $s$ is surjective is equivalent to prove $s(X)$ is both open and closed in $\overline{X}$.   Let $\bar{x}=s(x)\in \overline{X}$ for some $x\in X$.  As $(p\circ s )(x)=x$ we have $p(\bar{x})=x$.   For this $x\in X$ there exists an open set $U$ in $\overline{X}$ containing $\bar{x}$ such that $U\rightarrow p(U)$ is a homeomorphism.  As $U$ is open in $\overline{X}$ containing $\bar{x}$ and $s$ is continuous we have open set $V$ in $X$ with $x\in V$ and $s(V)\subset U$.  We have $$(p\circ s)(V)=V\Rightarrow (p^{-1}\circ p\circ s)(V)=p^{-1}(V)\Rightarrow (p^{-1}\circ p)(s(V))=p^{-1}(V).$$  As $s(V)\subset U$ the function $p^{-1}\circ p$ is identity on $s(V)$ so we have $s(V)=p^{-1}(V)$ So, $s(V)$ is an open set. we have $\bar{x}=s(x)\in s(V)\subset s(X)$. Thus $s(X)$ is open in $\overline{X}$.  I am not able to see why $s(X)$ is closed in $\overline{X}$.  This also says $p$ is also a homeomorphism.
If $q: E\rightarrow X$ is a covering map that has a section (i.e. $f: X\rightarrow E, q\circ  f=Id_X$) does that imply that $E$ is a $1$-fold cover?
if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this Book ID: Astronauts and Lab Students can create portals through space Desperate to locate and re-read exceptional book/series, but cannot remember the title. The high concept: The U.S. has just sent the first successful expedition to Mars, and it has just landed. As the astronauts make their initial exploration outside their ship, a tear in the fabric of space opens up right in front of them, and they look through to the insides of a laboratory (at a California university?) and two students who just figured out how to create arbitrary portals between two points in the Universe. The rest of the novel/series is a consequence of this breakthrough. But I can't remember the author and title(s). Surely someone reading this forum will remember it well! TIA.  (Reposted .) I have the following code:  MemoryStream foo(){     MemoryStream ms = new MemoryStream();     // write stuff to ms     return ms; }  void bar(){     MemoryStream ms2 = foo();     // do stuff with ms2     return; }   Is there any chance that the MemoryStream that I've allocated will somehow fail to be disposed of later?    I've got a peer review insisting that I manually close this, and I can't find the information to tell if he has a valid point or not. Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$. I have the following problem:      Let $(X,\tau)$ be a topological space and $(Y,\tau_y)$ be its    subspace, that is $Y\subset X$ and $\tau_y=\{\;A\;\cap\;Y\mid  A\in\tau\;\}$. Let also $F\subset Y$ and $H\subset X$, where $H$ is    closed in $X$. Show that $F=Y\;\cap\;H$ if and only if $F$ is closed   in $Y$.   I solved this problem by assuming that if $F$ is closed then $F=Y\;\cap\;H$, where $H$ is closed in $X$, but I got stuck proving this in the other direction, that is showing that if $F=Y\;\cap\;H$ then $F$ is closed in $Y$.  I tried to do something like this:   Assumption: $F=Y\;\cap\;H$, where $H$ is closed in $X$. If $H$ is closed in $X$ then $X\;\setminus\;H$ is open in $X$ and then $X\;\setminus\;H \in \tau$ which means that $B = (X\;\setminus\;H)\;\cap\;Y\in \tau_y$.  How should I proceed now? Should I investigate the set $B$ now more or what? Imagine a battery with the positive terminal connected to ground.  The positive side has a high eletrical potential, ground obviously has a lower electric potential.  Electricity wants to seek a lower potential, and there's no significant resistance between the terminal and the ground.  Shouldn't electricity flow?  I know it doesn't but I don't understand why.  I know you have to close the loop, but why is this necessary? I am a Indian citizen in the United States on a F-1 visa.  For the summer break, I am traveling (round-trip) from Chicago to New Delhi via Dusseldorf-Frankfurt by Lufthansa airlines. My onward Journey would be on July - 8 and return Journey on Aug - 11.  My layover will be of 1 hour 10 mins in Frankfurt and 3 hours in Dusseldorf.  I would like to know if I require Category A or Category B Visa. If not, what type of visa do I require for the Schengen visa for my transit. How I can apply for the same?  I tried to find slots in German consulate but all slots are full before my departure date. If I don't get the Schengen visa, I cannot travel and will loose all the fare (since my ticket is a non-refundable ticket). Is there any other way I can tackle this situation. Using InDesign CS6, how can I simultaneously edit A4 and A5 versions of a poster? I'm on Adobe Creative Cloud, so I have the latest InDesign, and I'm attempting to design a poster for my choir's next concert. Traditionally, I've done this in Photoshop, in two separate files, however I've recently started using InDesign after converting to it to create wedding albums through Blurb.  I've created an A4 document, and created an alternate-layout for it in A5. I've then lined them up side-by-side, and I've added a black frame to the A4 page, to act as the black background for my page (by the way, feel free to advise on better ways to do this).  My assumption is that I've failed to understand how this works, as I was hoping that I'd magically get a black frame on the A5 version, and that I could use liquid layout to get this to automatically scale (along with all other elements), in order to create the two versions simultaneously.  All help/advice would be greatly appreciated! Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? How to test real network throughput between two points? What are some of the better tools/utilities for testing real bandwidth across a link?  In my case I am testing the real throughput across a wifi bridge. What are the restrictions on the use of Slayer's Prey? In the recent book Xanathar's Guide to Everything, there are some subclasses/archetypes added for Rangers, among them the  (p. 43), which gives some specific helpful features. Hunter's Sense  At 3rd level, you gain the ability to peer at a creature and magically discern how best to hurt it. As an action, choose one creature you can see within 60 feet of you. You immediately learn whether the creature has any damage immunities, resistances, or vulnerabilities and what they are. If the creature is hidden from divination magic, you sense that it has no damage immunities, resistances, or vulnerabilities. You can use this feature a number of times equal to your Wisdom modifier (minimum of once). You regain all expended uses of it when you finish a long rest.  The limitations on use are clearly spelled out. Slayer's Prey  Starting at 3rd level, you can focus your ire on one foe, increasing the harm you inflict on it. As a bonus action, you designate one creature you can see within 60 feet of you as the target of this feature. The first time each turn that you hit that target with a weapon attack, it takes an extra 1d6 damage from the weapon. This benefit lasts until you finish a short or long rest. It ends early if you designate a different creature.  So from this, I would interpret it as it being able to switch from creature to creature at will, but there is no guidance as to what happens if the creature under the mark dies, or if you can carry it (so to speak) from encounter to encounter. Is there a limitation on the use of Slayer's Prey in the wording that I am missing? Or is this a basically a multiple use feature? (i.e., you can designate any creature as a bonus action, with no limitations on uses per day)? The extra d6 seems a little overpowering at lower levels, and somewhat meh at higher levels. FWIW, the GM basically allowed it as multi-use for the session, and was going to also try to find clarification. JavaScript: getting ImageData without canvas Detecting constexpr with SFINAE The identification of an electron as a particle and the positron as an antiparticle is a matter of convention. We see lots of electrons around us so they become the normal particle and the rare and unusual positrons become the .  My question is, when you have made the choice of the electron and positron as particle and anti-particle does this automatically identify every other particle (every other fermion?) as normal or anti?  For example the proton is a particle, or rather the quarks inside are. By considering the interactions of an electron with a quark inside a proton can we find something, e.g. a conserved quantity, that naturally identifies that quark as a particle rather than an antiparticle? Or do we also just have to extend our convention so say that a proton is a particle rather than an antiparticle? To complete the family I guess the same question would apply to the neutrinos. How can I change what keys on my keyboard do? (How can I create custom keyboard commands/shortcuts?) I want to change the key bindings on some of the keys on my keyboard.  I want some to run commands and others to activate different keys.  What applications can I use to do this?      Index:       (from within System Settings)             I'm still looking for answers using dconf and any other method. Could $X^3 + X + 1$ have a root in $E?$ Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 =$ {0, 1} with a root $α$ of $g.$  Could $X^3 + X + 1$ have a root in $E?$  When working through this sheet Ive come across this question in which im not sure how to answer so any help will be appreciated How can i find the python triggered by the GUI? How to do integer & float calculations, in bash or other languages/frameworks? I am trying to make a histogram with the fitted curve plotted on it. Here's the mwe:   \documentclass[a4paper]{scrartcl}     \usepackage{tikz}     \usepackage{pgfplots,pgfplotstable}     \usepackage{filecontents}     \pgfplotsset{compat=1.10}     \usepackage[active,tightpage]{preview}     \setlength\PreviewBorder{2pt}     \newcommand{\plots}{0.611201}     \newcommand{\plotm}{2.19882}     \begin{filecontents}{data.dat}     2 0.0629921259843     3 0.0236220472441     4 0.0314960629921     5 0.125984251969     6 0.0629921259843     7 0.102362204724     8 0.110236220472     9 0.0551181102362     10 0.0629921259843     11 0.0314960629921     12 0.0236220472441     13 0.0314960629921     14 0.0629921259843     15 0.0551181102362     16 0.0393700787402     17 0.0472440944882     18 0.00787401574803     19 0.0393700787402     24 0.00787401574803     27 0.00787401574803     33 0.00787401574803     \end{filecontents}     \begin{document}     \begin{preview}      \begin{tikzpicture}     \begin{axis}[     yticklabel style={/pgf/number format/fixed},     scaled y ticks = false,     minor y tick num={1},     xtick pos=left,     legend cell align = left,     legend style={draw=none},     xlabel = {group size},     ylabel = {ratio}     ]     \addplot[blue,ybar,fill, fill opacity=0.3, bar width = 0.8,] table {data.dat};     \addplot[red, line width = 1,domain=1:40,samples=100] {1/(x*sqrt(2*pi)*\plots)*exp(-(ln(x)-\plotm)^2/(2*\plots^2))};     \legend{empirical,lognormal fit}     \end{axis}     \end{tikzpicture}     \end{preview}     \end{document}     As you can see, the legend is not really good. How can I change the legend so it looks more like this? How do I capture the moon and its surrounding context? I wanted to photograph the full moon against a beautiful deep blue sky. I ended up with blown-out highlights of the moon:    So I switched to spot metering and captured the moon, losing the blue sky in the process:    I want a single photo that shows both the moon and the surrounding context (in this case, the blue sky). Doing an exposure fusion in Photomatix Essentials did not help:    Nor did HDR fusion (again in Photomatix):    Notice that the detail in the moon was lost in both cases, and the sky was also messed up in both cases: exposure fusion lost the beautiful blue color, while HDR created more texture than was actually present in the sky. I toggled Photomatix's option to remove ghosts, but that didn't help, either.  How do I photograph the moon together with its surrounding context (in this case, the blue sky, but in other cases, a tree, buildings, etc) without blowing highlights or shadows? I'm using a Sony NEX-5R, with the longest focal length lens I have, and with manual focus when needed.  Thinking I should fuse the photos manually, I tried opening the images as layers in Acorn, with the darker photo on top and with opacity set to 50%, and tried all 15 - 20 blending modes (normal, dodge, lighter, darker, multiply and so on), but none of them seemed to work. I'm afraid I don't know enough to use layers effectively. What blending mode and opacity and order of layers should I use for this task?  I don't have Photoshop, but do have Lightroom 5, Acorn and Nik Collection.  (In case you are about to recommend software, please note my requirements: I use a Mac, I would be willing to pay $20-30, and I don't want to use command-line software.) Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges