This is a question following . I pictured all the points of:  $$C = \{x\times 0 \mid 0&lt;x&lt;1\}$$  as a subset of the ordered square. (the question asks me to find the closure)  I calculated its limit points, which are $[0,1)\times 1$, andthe points of $C$, which are $(0,1)\times 0$. In order to calculate the closure (which is what the exercise asks, I need to unite the limit points with the set). Brian told me that $\langle 1,0\rangle$ is also a limit point, but I can't see why:  Look at the image below:   If I take an open set around $\langle 1,0\rangle$, there's no intersection with $C$, which are the points that lie on the ground, that is, the points on $(0,1)\times 0$
I need to find the closure of these sets on the ordered square: $$A = \{(1/n)\times 0 | n\in \mathbb{Z}_+\}$$ $$B = \{(1-1/n)\times \frac{1}{2}| n\in \mathbb{Z}_+\}$$ $$C = \{x\times 0 | 0&lt;x&lt;1\}$$ $$D = \{x\times \frac{1}{2}| 0&lt;x&lt;1\}$$ $$E = \{\frac{1}{2}\times y | 0&lt;y&lt;1\}$$ I know that the closure is just the set itself united with its limit points. For $A$, the set of limit points, I imagine, are $\{&lt;1,0&gt;\}$, that is, the $x$ coordinate is $1$ and the $y$ is $0$. But  is says the reverse. What am I getting wrong? Following the same reasoning, $B' = \{&lt;1,\frac{1}{2}&gt;\}$ $C' = \{&lt;0,0&gt;,&lt;0,1&gt;\}$ What am I doing wrong?
Redirect parent window from an iframe action Everywhere I get to read statements like EM radiations and EM fields carry "energy as well as momentum " . I wonder if energy and momentum are inalienable . Is it possible for a field to carry only energy without having any momentum term ? I mean isn't it sufficient to say that fields or EM waves carry energy or Doesn't the ability to transfer energy, of the fields, automatically imply that they can change the momentum of another particle while transferring energy to them? How can I enable Hardmode? When disconnecting from WebSphere MQ with C# client TCP connections are still in CLOSE_WAIT status How to get ongoing notifications of unread texts? One of my family members texts me at night when they are having nightmares. I want to know when they text me at 4 am instead of sleeping through it.  Is there a way to make text notifications keep going off until read? Why doesn't backpropagation work when you initialize all the weight the same value (say 0.5), but works fine when given random numbers?   Shouldn't the algorithm calculate the error and work from there, despite the fact that the weights are initially the same? Light Presets .exr I would like to install light presets for Blender to automate my lightning in my scenes. I have some .exr files (from Greyscalegorilla) but I don't know how to do it on Blender to make it work. Can someone help me? Thank you If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded. Show that $f'(x)\to0$ as $x\to\infty$. (This is inspired by  to a different question) How can I get the list of all fields(API name,Label,type) of any sObject? I'm trying to build a Dynamic list(table) of records by selecting the Sobject and its fields.So I need to get all the fields with its API name,label,type.But I can't get the label and type.(Tried DescribeSObjectResult Class methods to get label,but it didn't help to get the label dynamically)  I tried it with the following code.  System.debug('Selection-----'+selectedObject); Map&lt;String, Schema.SObjectType&gt; gd = Schema.getGlobalDescribe();  Schema.SObjectType ctype = gd.get(selectedObject);  Map&lt;String, Schema.SobjectField&gt; fmap = ctype.getDescribe().fields.getMap();    List&lt;FieldWrap&gt; strList = new List&lt;WrapMap&gt;(); for(String s: fmap.keySet()) {     WrapMap wmp = new WrapMap();     wmp.name = s;     wmp.label = String.valueOf(fmap.get(s));     strList.add(wmp);  }   Where FieldWrap is my wrapper class with name and field of String types. I'm getting the List of API name of fields,but not the Label and type of it.Is there a way to get the label and type of field.If so, please help me. I'm new to Delphi, and I've been running some tests to see what object variables and stack variables are initialized to by default:  TInstanceVariables = class   fBoolean: boolean; // always starts off as false   fInteger: integer; // always starts off as zero   fObject: TObject; // always starts off as nil end;   This is the behaviour I'm used to from other languages, but I'm wondering if it's safe to rely on it in Delphi?  For example, I'm wondering if it might depend on a compiler setting, or perhaps work differently on different machines.  Is it normal to rely on default initialized values for objects, or do you explicitly set all instance variables in the constructor?  As for stack (procedure-level) variables, my tests are showing that unitialized booleans are true, unitialized integers are 2129993264, and uninialized objects are just invalid pointers (i.e. not nil).  I'm guessing the norm is to always set procedure-level variables before accessing them? Why do corticosteroids harm COVID-19 patients? states that steroids could do more harm than good in treating coronavirus (COVID-19), referring to .      Understanding the evidence for harm or benefit from corticosteroids in   2019-nCoV is of immediate clinical importance.   It's unclear to me how exactly the use of corticosteroids (which is a common treatments for Asthma patients) can do harm other than the vague explanation of "steroids also impair the immune system's ability to fight viruses".   How exactly does the use of steroids harm the body in case of fighting a virus such as SARS-CoV-2 (2019-nCoV) that causes COVID-19? I'm studying special relativity and saw the 2 postulates of Einstein. The most remarkable one for me is the universal speed of light. Einstein postulated that the speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source. But what if we are accelerating and we are changing from one inertial frame to another, how will we observe the speed of light? Is it still the same? I suggest it's not? Because I alread saw a small introduction to general relativity where the light will bend in a gravitational field and therefore also when you accelerate. So does the speed of light change because it bends? Why do we multiply the magnetic flux by the winding number? $\mathbb Q$ is not the direct product of two non-trivial groups. I know this has been asked before, in a bunch of places, so I expect it to be closed as a duplicate at some point, but I'd like to know if my own proof works. If the proof is incorrect, please limit answers to pointing out the errors. If the proof is correct, then thoughts on improving it would be appreciated.  Prove that $\Bbb Q$ is not the direct product of two non-trivial groups.  —Aluffi exercise II.3.5. My proof Suppose that $\Bbb Q = G \times H$, neither $G$ nor $H$ trivial. First note that $0_G$ is the only element of $G$ with finite order: If $m\ne 0$ and $m\cdot g=0_G$, then $m\cdot(g,0_H)=0_{G\times H}$, contradicting the fact that $\Bbb Q$ has no non-zero elements of finite order. Suppose that for some $m\ne 0$ and some $n\ne 0$, $\pi_G(m/n)=0_G$. Then $0_G=n\cdot\pi_G\left(\frac m n\right)=\pi_G\left(n\cdot\frac m n\right)=\pi_G(m)=m\cdot\pi_G(1)$. Since $\pi_G(1)$ has finite order, we conclude that $\pi_G(1)=0$. Since $\Bbb Z=\langle 1\rangle$, $\pi_G$ maps every integer to $0_G$. Now let $a$ and $b$ be non-zero integers. Then $b\cdot\pi_G\left(\frac a b\right)=\pi_G\left(b\cdot \frac a b\right)=\pi_G(a)=0_G$. As above, this shows that $\pi_G\left(\frac a b\right)=0_G$. Thus if $\pi_G$ is not an injective function, then it must map everything to $0_G$, so $G$ will be a trivial group. The same result holds for $\pi_H$. Since by assumption, $G$ and $H$ are non-trivial, $\pi_G$ and $\pi_H$ are both injective functions, which is of course impossible. How can I remove all unused materials from a .blend file? Why won't Blender run on Windows XP? I downloaded Blender 64 bit and the OS indicated that the program was not compatible with my computer chip. I then downloaded the 32 bit version which seemed to allow me to install. However the blender Icon in windows when pressed informs me that the program is not a window application. I have an Intel(R) Core (TM)2 Duo CPU E7300 @2.66Ghz if that is important and I am running XP. I am fairly sure I am doing something wrong here- can anyone help please? Question, is it ok for one of the pages levels to 404? (SEO-wise)  Let's say I have a sitemap like this:  example.com --/services   --/foo   --/bar --/sectors   --/my-sector   --/another-sector   --/the-sector   --/stuff --/about-example --/contact --/etc --/etc   This makes a lot of structural sense. However, from a design point of view, I think having a page to show all the sectors (example.com/sectors) is pointless since all of the sector pages can be accessed from the home page, main navigation, and in each sector page.  Would there be any SEO consequences if the /sectors page were to 404?  Note: in the sitemap 'example.com/sectors' would not be included. Different colors for different types of blocks in beamer In a beamer presentation, I’d like to set up different colors for normal blocks, definition blocks, theorem blocks, proof blocks, example blocks…  Ideally I’d set those up using \usebeamercolor commands. I can do it for example blocks, and for all other kinds of blocks at once, as illustrated in the sample below (compiled using XeLaTeX). But I can’t find how to do it for specific, non-example, types of blocks.  I suppose “” points towards some solution, but I’d prefer (somehow) inheriting from the existing special block environment definitions rather than define new environments from scratch.  \documentclass{beamer} \usepackage{fontspec}  \useinnertheme[shadow=true]{rounded}  \setbeamercolor{block title}{use=structure,fg=structure.fg,bg=structure.fg!20!bg} \setbeamercolor{block body}{parent=normal text,use=block title,bg=block title.bg!50!bg}  \setbeamercolor{block title example}{use=example text,fg=example text.fg,bg=example text.fg!20!bg} \setbeamercolor{block body example}{parent=normal text,use=block title example,bg=block title example.bg!50!bg}  \begin{document} \begin{frame}     \begin{block}{My block}         A block.     \end{block}     \begin{definition}[My definition]         A definition.     \end{definition}     \begin{theorem}[My theorem]         A theorem.     \end{theorem}     \begin{proof}[My proof]         A proof.     \end{proof}     \begin{example}[My example]         An example.     \end{example} \end{frame} \end{document} How can I upgrade from Mac OS X Tiger 10.4? If I buy a regular retail Snow Leopard disc (i.e. the £25 version), can I use it to upgrade a Mac from Tiger (10.4)?   on the Apple Store’s Q&amp;A site suggests it’ll work, but Apple seem to imply that you need the (significantly more expensive) Mac Box Set (“”). Recently I have installed an older version of Ubuntu on my old machine. Whenever I try to install any software, I get an error saying it couldn't be found:  $ sudo apt-get install vlc Reading package lists... Done                Building dependency tree        Reading state information... Done     E: Couldn't find package vlc