$\emptyset$ is called both open and closed. Trying to understand this. It's complement is $R$ for instance. Does that mean R is both open and closed? R is unbounded for sure, but every cluster point of R is contained in R (which satisfies the definition of being closed). R is closed looked from this point of view. This would make $$\emptyset$$ open.  But R could also be considered as open, since every point $p_o$ has a neighbourhood which is interior to $R$. Is this why $\emptyset$ is considered both open and closed, therefore $R$ is considered both open and closed?  Another example :  Considered the set $$S={x^2+y^2≥0}$$ Can this be considered both open and closed then?. This condition is satisfied for all $x,y$ therefore this set would be the $R$ itself. Complement of this set is the empty set, and since empty set is considered both open and closed as proved above. This set is also considered both open and closed.
I think this is a basic question, but it's hard to wrap my head around.
I already have an Apple ID, made at Apple.com. Now, when I want to access the App Store from my iPad, I am told that I have to review my information. On the credit card page, I am not presented with a "None" option, even though I am in the United States.   What can I do to proceed without entering this information? Whenever I try to install anything with the Ubuntu Software Center, the following error appears.     Failed to download repository information      Check your Internet connection.      E:Unable to parse package file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_precise_multiverse_i18n_Index (1) strtok segmentation fault How to understand Cauchy's proof of AM-GM inequality(the last step) What is the product of the empty set? Give:  $fn(S)=\prod_{x\in S}x$  what is:  $fn(\emptyset)$  I can see reason that it would be defined as 1 or 0.    Note: I thought about restricting the domain of $S$ but that would make the problem less general. If there is no general answer then answers for restricted domains would be valid. Why do only two sexes exist for animals? Why, from the natural selection point of view, do only two sexes exist for animals? Can it hurt to put my GRE score on my resume? Induction proof: $S$ contains powers of 2 and predecessors implies $S={\bf N}$ Let $S$ be a subset of $\mathbb{N}$ such that $$2^k\in S\quad\forall\ k\in\mathbb N$$ and, $$k\in S\Rightarrow k-1\in S\quad\forall\ k\ge2,k\in\mathbb N$$  Show by induction that $S=\mathbb N$. Reference undefined in \section with amsbook I have a section that is devoted to the proof of a theorem, and hence I used  \section{Proof of Theorem~\ref{thm}}   Of course, because of the headers that are capitalized, it asks for \ref{THM} when producing the header, and I get an obvious error. What is the proper way to handle such a case? Below is a MWE:  \documentclass{amsbook} \usepackage{lipsum} \newtheorem{thm}{Theorem} \begin{document} \chapter{First Chapter} \section{The Theorem} \begin{thm}[Lipsum Theorem] \label{thm} \lipsum[1] \end{thm} \section{Proof of Theorem~\ref{thm}} \lipsum[1-23] \end{document}   When compiling I get:  LaTeX Warning: Reference `THM' on page 3 undefined on input line 14.   (each time it produces a header), and the header shows     `2. PROOF OF THEOREM ??' By default after a new user signs up, a redirect to the home page happens.  But in such case I want to redirect such user to another page like node/35.  In this node/35 I then want to display some content. I've tried to construct examples of rings that match all except one of the properties in the definition of a Dedekind domain. (This is an old number theory qual question from Berkeleys MGSA website).  The only starting point that I can think of would be  $R:=K[X_1,X_2,\ldots]$  for $K$ a field. This is clearly integrally closed as any polynomial relation is contained in $K(X_1,\ldots,X_n)$ for some large enough $n$ and $K[X_1,\ldots,X_n]$ is integrally closed being a UFD.  The problem is then to get rid of enough ideals from this ring $R$, so that its dimension would be 1, but I can't seem to figure out what to do.  Anyone have any other ideas for rings that could work as a starting point? The polynomial ring with an infinite number of variables is pretty much the only example that I know of for how to construct a non-noetherian ring that's still a domain. How do I choose an element from a non-empty set? Suppose I have a non-empty set $A$.     How do I choose an element $x\in A$?   More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is exactly one $x\in y$ such that $P(x,y)$ is true.  I have thought about this for a while without success and I am beginning to doubt such a formula is even possible. But in mathematical proofs it is quite common to "choose a fixed element" from a non-empty set. So how exactly do we achieve such a thing? Am I missing something?  Thank you in advance. What's the relationship between initial eigenvalues and sums of squared loadings in factor analysis? On the one hand I read in a comment  that:     You can't speak of "eigenvalues" after rotation, even orthogonal   rotation. Perhaps you mean sum of squared loadings for a principal   component, after rotation. When rotation is oblique, this sum of   squares tells nothing about the amount of variance explained, because   components aren't orthogonal anymore. So, you shouldn't report any   percentage of variance explained.   On the other hand, I sometimes read in books people saying things like:     The eigenvalues associated with each factor represent the variance   explained by that particular factor; SPSS also displays the eigenvalue   in terms of the percentage of variance explained (so factor 1 explains   31.696% of total variance). The first few factors explain relatively large amounts of variance (especially factor 1), whereas subsequent   factors explain only small amounts of variance. SPSS then extracts all   factors with eigenvalues greater than 1, which leaves us with four   factors. The eigenvalues associated with these factors are again   displayed (and the percentage of variance explained) in the columns   labelled Extraction Sums of Squared Loadings.   That text is from Field (2013) Discovering statistics using IBM SPSS, and this diagram accompanies it.    I'm wondering     Who is correct about whether it's possible to speak of eigenvalues after rotation? Would it matter if it was an oblique or orthogonal rotation?   Why are the "initial eigenvalues" different from the "extraction sums of squared loadings"? Which is a better measure of total variation explained by the factors (or principal components or whatever method is used)? Should I say that the first four factors explain 50.317% of variation, or 40.477%? Do cards with abilities that target trigger heroic? As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that? How to setup Squid Proxy (for localhost services and Tor and I2P) Getting Started With Java EE I am about to start a web project and have been working almost exclusively with a LAMP stack for the past 5 years.  I would like to build something larger and more scalable and hence have been looking into Java EE.  The division of logic between JSP and EJB seems ideal for what I want to do.  Can anyone recommend resources for getting familiar with Java EE?  And perhaps if anyone knows of technologies that are more scalable than PHP that I should consider as well, I'm all ears.  Thanks a lot! How can I randomise the output of a redstone circuit? I have 5 buttons and only one of these can be correct, all others must be false. These 5 outputs will go into 2 outputs, in between there needs to be a circuit which changes which single button out of the 5 will go down the &quot;correct&quot; output of the final 2. The 4 remaining buttons must go to the second of the final 2 outputs. Both final outputs must give out a true signal for a brief period the same as a button on its own would produce. There should also be a way of activating the circuit. Not sure if that is explained well enough so I have made a basic diagram of what I need.  The diagram shows how I need the signal to be split. However in between the buttons and the final two wires there needs to be a circuit (marked by the grey box). This circuit changes the single button connected to the left wire. The right input should start the circuit and randomise the 5 buttons. There are no limits to size, amount of materials used or method used. This question is similar to the one asking about , but I am specifically working with Redstone circuitry - the commands solutions do not apply here Due to my reputation on Stack Overflow, I have access to different review queues. I want to know if it will change my score/reputation or if it will only provide me some badges?  Maybe this part seems a bit odd, but I want to know why some users spend so much time in the review queues? Is it something about their passion about Stack Overflow or their passion to help others inside the community or does it have some other reasons which I've missed?  P.S. I have read some information like  and  but I will be grateful to have your answer about this. How do I set a monitor resolution that is not available in the Display Settings dialog?