Let $X$ be an arbitrary set. The non-empty open nsets are the complements offinite sets. We have to define also the empty set. A sequence $x_n \to x$ converges as for the cofinite topology iff for each open neighbourhood $U$ of $x$, $U = X \setminus \{s\}$ for a $s$ with $x \neq s$, it holds that almost all $x_n$ are in $U$. So if $x \neq s$, thenalmost all $x_n \neq s$. If infinitelymany $x_n=s$ then $x=s$. Have I understood that correctly? So considering the case $x_n=\frac{1}{n}$: Let $U$ be an open neighbourhood of $0$. Since $\Bbb R\setminus U$ must be finite, $U$ containsallbut a fininte numberof terms of the sequence. Therefore $x_n\rightarrow 0$. Is that correct?
I consider the co-finite topology on $\mathbb{R}$ that is $$\{G\subset \mathbb{R}, card(\mathbb{R}\setminus G)&lt;+\infty\}\cup \{\emptyset\}$$  What are the convergent sequences in this space .  thank you
What is the DC of the Tempest Cleric's Wrath of the Storm feature? Is -1 followed by a singular or plural noun? Do you use "a" or "an" before acronyms / initialisms? Does every Abelian group admit a ring structure? Is this a correct demonstration for why elements above untriseptium cannot exist? With the confirmation that elements 113, 115, 117, and 118 are indeed fundamental elements that are now to be named on the periodic table, the next question is: what is the highest atomic number possible for an element? Feynman had a go at this years ago, and he derived (according to my limited understanding) that above element-137 (informally denoted 'Feynmanium') the electrons in the nearest orbit about the nucleus would be traveling at a velocity greater than the speed of light. (Note I am considering the Bohr model of the atom in this question.) I wished to demonstrate to my friends why this is and I came up with the following explanation.  According to Bohr's quantum condition, the angular momentum of an electron about the nucleus is given by $$ L=m_evr_n=n\hslash \implies r_n=\frac{n\hslash}{m_ev},$$ where $m_e$ is the mass of the electron, $v$ its velocity, $n$ an integer, and $r_n$ the radius of the $n$th possible orbit. Because we are concerned with the nearest orbit to the nucleus, $n=1$; and thus $$r=\frac{\hslash}{m_ev}.$$   Now, according to Coulomb's law, if the electron orbits about the nucleus the centripetal motion can be described by $$\frac{Ze^2}{4\pi\epsilon_or}=m_ev^2, $$ where $Z$ denotes the number of protons in the nucleus (the atomic number), and $e$ the elementary charge. Solving for $Z$ and substituting in $r$ from above yields $$Z=\frac{4\pi\epsilon_o \hslash v}{e^2}. $$ But what is $v$? Well, the maximum velocity an electron could ever have is the speed of light, and we wish to find the atomic number associated with an orbiting electron traveling at this speed, so we set $v=c$ and obtain our final result of $$Z=\frac{4\pi\epsilon_o\hslash c}{e^2}\approx 137.521,$$ which implies that for $Z&gt;137$, the electrons at a position of $n=1$ in the Bohr model would have a velocity $&gt;c$; and thus the highest atomic number achievable on the periodic table is 137.  Again, I just want to make sure this is a correct method for deriving element-137 before presenting. Perhaps one could explain how relativity plays a role here. I know Feynman used the Dirac equation to get this result...so could anyone (subsequently of course) expatiate on this in a simplistic manner? Thanks! Polynomial bounded real part of an entire function Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial?  Or, without loss of generality and more suggestively $$(f(z)=\sum_{k=0}^\infty a_k z^k \quad \land \quad \operatorname{Re}(f(z))\leq |z|^n)\quad \forall z\in \mathbb{C} \qquad \Rightarrow \qquad f(z)=\sum_{k=0}^n a_k z^k \quad\forall z\in \mathbb{C}$$ I want to shade the area of overlap for adjacent normal curves, shown in the lower graph. I can get shading between the first two curves but not for subsequent curves. I need shading at overlapping areas indicated by the arrows. I applied the "logic" of the first fill for the subsequent curves but that does not work.   The arrows and letters in the example image are not part of the MWE.   \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{ticks=none} \pgfplotsset{compat=1.7}  \usepgfplotslibrary{fillbetween}  \pgfmathdeclarefunction{gauss}{2}{%normal distribution where #1 = mean and #2 = sd}     \pgfmathparse{exp(-((x-#1)^2)/(2*#2^2))}% }  \pgfplotsset{baseplot/.style={%     no markers,      domain=1:4.5,      samples=100,     smooth,     axis lines*=left,     height=5cm, width=12cm,     enlargelimits=upper, clip=false, axis on top,     xlabel = near ticks,     xlabel={Resource} }}  \begin{document}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(2.7,0.4)};   \addplot+[name path=B]{gauss(3.2,0.4)};   \fill[gray!20, intersection segments ={of= B and A}]; \end{axis} \end{tikzpicture}  \vspace*{3\baselineskip}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(1.7,0.15)};   \addplot+[name path=B]{gauss(2.2, 0.15)};   \addplot+[name path=C]{gauss(2.7, 0.15)};   \addplot+[name path=D]{gauss(3.2, 0.15)};   \addplot+[name path=E]{gauss(3.7, 0.15)};   \fill[gray!20, intersection segments ={of= B and A}];   \fill[gray!20, intersection segments ={of= C and B}];   \fill[gray!20, intersection segments ={of= D and C}];   \fill[gray!20, intersection segments ={of= E and D}]; \end{axis} \end{tikzpicture}  \end{document} How to repair Ubuntu installation after interrupted dist-upgrade without losing any custom settings? Storing hard crusted bread I'm currently baking a batch of sourdough bread. My last batch turned out perfectly with a hard crust and chewy inside. My problem is how to store the bread such that the crust stays hard. If I put it in a plastic bag the crust goes soft in a matter of hours and I'm afraid that the bread will turn dry if I leave it without some kind of protection. Note that I'm not looking for a long term storage solution.  What is the best way to store my bread, such that the crust stays hard and the inside stays chewy? Is it possible to call the base method from a prototype method in JavaScript if it's been overridden?  MyClass = function(name){     this.name = name;     this.do = function() {         //do somthing      } };  MyClass.prototype.do = function() {       if (this.name === 'something') {         //do something new     } else {         //CALL BASE METHOD     } }; I'm using iPhone 6s 16 GB. My storage isn't full but still I can't download anything. Every time I wanted to start a download it keeps telling:  app name* could not be download at this time How do I suppress the "New release '12.10' available" message? How to make existing public files private? What is the mechanism behind the slowdown of light/photons in a transparent medium? So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the "vacuum" between the atoms? How can I cast int to enum? MS SQL Server: Do multiple queries in a batch ever execute in parallel and if so what happens when the second query is dependent on the first? Does cinnamon 2.0 really break your 13.10 desktop? As title,  Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?  Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, this implies that $K$ is finitely generated (as an algebra) over $F$, since a basis is a generating set. So every finite extension is finitely generated.  So indeed they all are, is my logic correct? iMac only show flashing folder with question mark after formatting OS X partition drive Today i format my "OS X El capitan" partition for installing "Yosemite". But due an error in bootable yosemite usb it didn't install on this partition. After some hours when i turned on iMac and hold down "option" key for boot into usb but just blank screen and mouse show. And every time when i turend on iMac it only show flashing folder with question mark. I've tried to resolve it by reseting "NVRAM". But after restarting problem still same.  Also I've tried many "Startup key combinations for Mac" but didn't work any combination on startup.  Also i have important data in my other one partition. Please can anyone tell me what i do for resolving that issue Without loosing my data? How do I set up SIP voice calling in Empathy? I would like to use Empathy for Internet voice chat, but I do not see an option under the Empathy Accounts menu to create a SIP account.  How do I set up Internet voice calling in Empathy?