Citation undefined
Question mark or bold citation key instead of citation number
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then   $0$ is an isolated point $(-2,0]$ is an open set $0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$ $(-2,0)$ is open set.   $0$ is not an isolated point of $X$ as every nbd of $0$ contains a point of the form $1\over n$ so  $3$ is true and $1$ is false, $2$ is false as $0$ is not interior point, $4$ is true as every point is interior point. Am i right? Feedback Request: New Timeline Question View Why do professional print houses use gamut-limiting CMYK? While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? How to test multiple variables against a single value? I am travelling to Athens, Greece, shortly, and have seen . Should I be worried? Should I cancel my trip?  Update  It's fine, taxis are currently on strike, but Syntagma Sqaure is fine: PHP mysqli_fetch_field data type Mint 18 will boot only to a black screen How do you access the matched groups in a JavaScript regular expression? Book with travellers, alien artefacts and age reversal Looking for the title of a book I read about 5 years ago though I believe the book was a lot older.  Essentially the main male character was on a task and was travelling with a group of people. The mode of transport if I recall was a boat and they were on a remote planet travelling via a river. One of the group was a father to a young female infant. There's a reference to alien artefact/temple that essentially reverse aged a female scientist (the baby) a day at a time and the father is trying to undo it. Also a line I remember is the father used to say see you later alligator and the daughter will reply that awhile crocodile until a day where she forgot.  Another fellow traveller was a pilgrim and his male disciple. There are several other male and female travellers all had different reasons to travel to that location.  The aliens was located within the temple (time locked?) Were invisible to humans. Killed any that entered. Got the impression that they were quadpeds with organic blades/razors that cut people.  I believe the book may be part of a series. The male author though I don't recall the name, has written other works that have received awards. The book or series may have received one too. Which is probably why I decided to read it in the first place.  Question never asked anywhere else. I am from Australia, book was a paper copy bought locally. How to determine function name from inside a function How to do integer & float calculations, in bash or other languages/frameworks? Using echo "20+5" literally produces the text "20+5".  What command can I use to get the numeric sum, 25 in this case?  Also, what's the easiest way to do it just using bash for floating point? For example, echo $((3224/3807.0)) prints 0 :(.  I am looking for answers using either the basic command shell ('command line') itself or through using languages that are available from the command line. I tried to learn Dvorak keyboard layout a few years ago, and got reasonably good at it, but I couldn't get past the fact that command-key shortcuts were no longer in the right place.  In particular, Ctrl+Z (undo), Ctrl+X (cut), Ctrl+C (copy), and Ctrl+V (paste) are all chosen specifically because of the location of those keys on a Qwerty keyboard (well, maybe Ctrl+C makes sense).  Ctrl+A (select all) and Ctrl+S (save) are also very convenient because of their Qwerty keyboard location.  Is there any way (on Windows XP) to use Dvorak except when a control key (Ctrl, Alt, and/or WinLogo) is pressed?  (Preferably without having to install any third party app.) I seem to have clicked on something that changed my display preferences. The main site is now very minimalistic, with fewer controls and different fonts.  I want to change it back, but where is it? Looking for story title - younger gen waters down age-slowing drink of older gen, crammed into small apartment I remember a short story in which multiple generations of people being crammed into a small apartment in a city. The magical drink that either reduced or completely stopped aging was something like a soda that came in a six pack. The story focused on visual aspects of the drink at some point. The younger family members complained that their elders were living too long and apparently decided to water down the soda, I think that conclusion was ultimately left up to the reader. Many thanks for any help tracking down this story, might be one of Kurt Vonnegut's perhaps? Flying EU to UK to EU. Can I get duty free alcohol? On the 20 of March this year I will be flying from Ireland to the UK and then on the another EU country.  Can I buy duty free alcohol (as in alcohol with no tax) leaving the UK ? I'm trying to evaluate $(\hat{a}+\hat{a}^\dagger)^k|n\rangle$ Where $\hat{a}$ and $\hat{a}^\dagger$ are ladder operators and $|n\rangle$ the $n$th Fock state. For this, I separated the problem in three parts: $k&lt;n$, $k=n$ and $k&gt;n$. In this part I'm a bit confused: Is it possible to evaluate such operations? Some tips on how to start would be really great. Adding feature class to Map in CURRENT project using ArcPy in ArcGIS Pro? default constructor for int Intersection and union of simply connected domain I have some basics concept that, in $\mathbb{C}$, a "simply connected domain $D$" is a region in $\mathbb{C}$ with no holes. I am not sure whether it has various formal definition , but the one used in my complex analysis class looks very complicated. I barely understand what it means :  $\textbf{Definition}$ Let $\gamma_0, \gamma_1$ be two continuous closed path in a domain $D \subseteq \mathbb{C}$ parameterized by $I = [0,1]$;that is, $\gamma_i(0) = \gamma_i(1)$ for all $i = 0, 1.$ We say $\gamma_0, \gamma_1$ are $\textbf{homotopic as closed paths on D}$ if there exists function $\delta : I \times I \rightarrow D$ such that   $1) \ \delta(t,0) = \gamma_0(t) \ \forall t \in I \ \ $    $2) \ \delta(t,1) = \gamma_1(t) \ \forall t \in I \ \ $     $3) \ \delta(0,u) = \delta(1,u) \ \forall u \in I.$  A continuous closed path is $\textbf{homotopic to a point}$ if it is homotopic to a constant path(as a closed path).  $\textbf{Definition}$ A region $D \subseteq \mathbb{C}$ is called $\textbf{simply conected}$ if every continuous closed path in $D$ is homotopic to a point in $D$.  $\textbf{Problem :}$ 1) Let $D_1, D_2$ be simply connected plane domains whose intersection is nonempty and connected. Prove that their intersection and union are both simply connected.  2) Let $P, Q$ be smooth functions on a domain $D \subseteq \mathbb{C}$, Find necessary and sufficient condition for the form $P dz + Q d\bar{z}$ to be closed.