Comparing box topology and product topology, there are many better theorems standing in product topology, but not stands in box topology. So is it useful to study box topology?
In point-set topology, one always learns about the : the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open.  This seems naively like a "good" topology to use for $X$.  However, one quickly learns that this is not so; that the  is the natural one.  The box topology has many strange properties that make it a good source for counterexamples, but I am not aware of it having any other applications.  So I would like to know:     Are there examples of using the box topology to prove interesting "positive" statements?   Edit: And to pursue a comment of Jim Conant:     Are there "non-artificial" problems where the box topology arises naturally?   Edit: The title is perhaps too flippant.  I don't mean to minimize the obvious significance of the box topology as a counterexample.  However, for the purposes of this question I am interested in positive results.  I'm not looking to be convinced that counterexamples are useful; I know that they are.
I know this question is someway similar to  where they recommend using 404, but after reading  where they suggest to use 301 when changing site URLs (in the specific case was due to redesign/re factoring) I get a bit of confused and I hope someone could clarify for this specific example:   Let's say I have an eCommerce site, let's also say the final user inserted some interesting items in the site and the eCommerce webapp created the item pages at the URLs: http://...?id=20, http://...?id=30 etc. Now let's say some of these interesting items got many external links toward them from many other sites because some people found those items very interesting and linked to them. After some years the final user deletes those items, so obviously the pages and URLs http://...?id=20, http://...?id=30, etc. now do not exist anymore, but still many pages on the web are linking toward them.   What should the eCommerce site do now, just show a 404 page for those items?  But, I'm confused, wouldn't this lose all the Google PR passed by the external links to the items pages? So isn't it better to use 301 redirect to HOME PAGE that at least passes the PR to the HOME PAGE?    EDIT:  Well, according to answers the best thing to do so far is to do a 404/410.  In order to make this question more complete, I would like to talk about a special case, just to make sure I understood. properly.  Let's say the user creates those items again (the ones he previously deleted at point 4), maybe he changes a bit their names and description, but they are basically the same items. The webapp has no way to know these new added items were the old items so it obviously create them as new items with new URLs http://...?id=100, http://...?id=101, does it makes sense at this point to redirect 301 the old urls to the new ones?    ANOTHER EDIT (It would be VERY IMPORTANT TO UNDERSTAND):  Well according to the clever answers received so far it seems for the special case, explained in my last EDIT, I could use 301, since it's something of not deceptive cause basically the new pages is a replacement for the old page in term of contents. This is basically done to keep the PR passed from external link and also for better user experience.  But beside the user experience, that is debatable (*1), in order to preserve PR from external broken links why not just always use 301, In my understanding Google dislikes duplicated contents, but are we sure that 301 redirect to HOME PAGE is seen as duplicated contents for Google?!   Google itself  so if they consider 301 as duplicated contents wouldn't that be considered duplicated contents too? Why do they suggest it?  Let me provoke you: “why not just add a 301 to HOME PAGE for every not found page?”   (*1) as a user, when I follow a broken URL from some external link to some website's page I would stick more on this website if I get redirected to HOME PAGE rather than seeing a 404 page where I would think the website does not even exist anymore and maybe I don't even try to go to HOME PAGE of the website. Matlab filter electical spikes in accelerometric data A film or TV series about classic monsters getting together If $H$ has maximum vertex degree at most $3$ and is a minor of $G$, then $H$ is a topological minor of $G$. If  $H$ has maximum vertex degree at most $3$ (so $\Delta(H)\leq 3$) and $H$ is a minor of $G$, then $H$ is a topological minor of $G$.   The converse follows by definition. However, most sources state this proposition without a proof. Any help is greatly appreciated. Setting print layout to portrait using QGIS 3? Accidental chown under / as root I accidentally run the following command as root chown mike -Rf / backups the point is that I added a space between / and backups my mistake, I stopped the command after 1-2 seconds, how can I recover from this? I have problems with DirectAdmin right now for example when accessing it I recive the error:  Unable to determine Usertype user.conf needs to be repaired http://help.directadmin.com/item.php?id=456   Or I stopped recieving emails, I dont even know what else is messed up right now... A sum of series problem: $\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$ I have a question regarding the sum of this series: $$\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$$ My approach: I found that this sum is equal to: $$\sum_{n=3}^{2008}\frac{n}{(n-2)!+(n-1)!+(n)!}$$  I reduced it to : $$\sum_{n=3}^{2008}\frac{1}{n(n-2)!}$$  Please suggest how to proceed further. Suppose there are several journals with the same impact factor. Is it better to submit my papers to one of them or each time I finished a manuscript it is better to submit to an equally good journal that I have not published in yet.  I am early in my career and I already published two papers in the same journal and I am going to finish my third paper, which potentially can be published in the same journal as my previous two papers. How to check whether a string contains a substring in JavaScript? Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. My trackpad is broken so I can tap-to-click but i cannot drag and drop. This is really a problem when it comes to doing certain things.   Does anyone know how to program a sticky key to turn a click into a click-select that can then be dragged about?   I can't resize my browser or open window and I can't drag applications into the folder when installing etc.  free solutions only please.  A mid 2009 macbook-pro 13". 2.53 Ghz Intel Core 2 Due.   Running os lion (grudgingly). The Definitive C++ Book Guide and List This question attempts to collect the few pearls among the dozens of bad C++ books that are published every year. Unlike many other programming languages, which are often picked up on the go from tutorials found on the Internet, few are able to quickly pick up C++ without studying a well-written C++ book. It is way too big and complex for doing this. In fact, it is so big and complex, that there are very many very bad C++ books out there. And we are not talking about bad style, but things like sporting glaringly obvious factual errors and promoting abysmally bad programming styles. Please edit the accepted answer to provide quality books and an approximate skill level — preferably after discussing your addition in . (The regulars might mercilessly undo your work if they disagree with a recommendation.) Add a short blurb/description about each book that you have personally read/benefited from. Feel free to debate quality, headings, etc. Books that meet the criteria will be added to the list.  Books that have reviews by the Association of C and C++ Users (ACCU) have links to the review. *Note: FAQs and other resources can be found in the  and under . I am Indian passport holder living in Hungary for 4 years and I have been issued a Temporary residence permit (EU Schengen).  Can I go to visit Dublin without any visa, or do I need to apply for a UK visa? I know Ireland still in Schengen but part of United Kingdom. Do particles ever touch each other during a collision? How can I reference the script element that loaded the javascript that is currently running?  Here's the situation. I have a "master" script being loaded high in the page, first thing under the HEAD tag.  &lt;!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"&gt; &lt;html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"&gt; &lt;head&gt; &lt;script type="text/javascript" src="scripts.js"&gt;&lt;/script&gt;   There is a script in "scripts.js" which needs to be able to do on-demand loading of other scripts. The normal method doesn't quite work for me because I need to add new scripts without referencing the HEAD tag, because the HEAD element hasn't finished rendering:  document.getElementsByTagName('head')[0].appendChild(v);   What I want to do is reference the script element that loaded the current script so that I can then append my new dynamically loaded script tags into the DOM after it.  &lt;script type="text/javascript" src="scripts.js"&gt;&lt;/script&gt; loaded by scripts.js--&gt;&lt;script type="text/javascript" src="new_script1.js"&gt;&lt;/script&gt; loaded by scripts.js --&gt;&lt;script type="text/javascript" src="new_script2.js"&gt;&lt;/script&gt; Getting 2-factor Gmail authentication code from Sweden mobile number I have been facing a serious problem for many months. Usually I get a 2-factor authentication code from 51501 when I am supposed to log into Gmail. But for months I am getting a code from a Sweden-based mobile number: +46701327805. The first code I received from this number was in October 2015.  There is also a mobile number from USA that sometimes sends the code. I change my passwords every time this happens. What could be the possible reason for this and how to overcome this problem?   By the way, I must say I have been hacked for years. I don't use VPN; my internet runs off a broadband modem. ffmpeg working from command line in Terminal but not in an Automator shell script! I have just compiled my own copy of ffmpeg on my system and it's located in the following path:  /opt/local/bin/ffmpeg   When I run the following command in Terminal.app, it runs smooth:  ffmpeg -i /Users/Amit/Documents/input.avi -c:v copy -c:a aac -b:a 384k -strict -2 /Users/Amit/Documents/output.m4v   However, a similar command run as a shell-script in an Automator service fails:  for f in "$@" do     ffmpeg -i "$f" -c:v copy -c:a aac -b:a 384k -strict -2 "${f%.*}.m4v" done   Up until now, I had a pre-compiled build of ffmpeg on my system located at:  /Applications/Scriptlets/   And my Automator shell-script had the full path (/Applications/Scriptlets/ffmpeg) instead of just ffmpeg; and it worked fine. However, today I decided to compile a fresh copy of ffmpeg using MacPorts and removed the earlier version from my Applications/Scriptlets folder. Now, this ffmpeg works fine in Terminal but gives an error whenever I try running it from a shell-script inside of an Automator workflow. Am I doing something wrong here? Please help! If you have two envelopes, and ... Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch to the other one?  Intuitively, you don't know anything about either envelope, so it'd be ridiculous to say that you should switch to the other envelope to maximize your expected money.   However, consider this argument. Let $x$ be the amount of money in the envelope you picked. If $y$ is the amount of money in the other envelope, then the expected value equals  $$E(y) = \frac{1}{2}\left(\frac{1}{2}x\right) + \frac{1}{2}\left(2x\right) = \frac{5}{4} x$$  But $5x/4 &gt; x$, so you should switch!    The Wikipedia article says that $x$ stands for two different things, so this reasoning doesn't work. I say this is not a valid resolution.   Consider opening up the envelope that you pick, and finding $\$10$ inside. Then you can run the expected value calculation to get $$E(y) = \frac{1}{2} \cdot \$5+\frac{1}{2} \cdot \$20 = \$12.50$$  This means that if you open one of the envelopes and find $\$10$, you should switch to the other envelope. The $\$10$ doesn't stand for two different things, it literally just means $\$10$.   But you don't have to open up the envelope to run this calculation, you can just imagine what's inside, and run the calculation based on that. This is what "Let $x$ be the amount in the envelope" means. The problem with the argument is not that $x$ stands for two different things.   So what is the problem?    Previous questions on stack exchange have given the resolution that I just said I wasn't satisfied by, so please don't mark this as a duplicate. I want a different resolution, or a more satisfying explanation of why $x$ does stand for two different things.  Apparently there is still research being published about this problem - maybe it isn't so obvious?    I think there's something subtle wrong with the premise. Because there's no uniform probability distribution on $\mathbb{R}$, statements like "random real number" are not well-defined. Likewise, I think "one envelope has twice as much money as the other" assumes some probability distribution on $\mathbb{R}$, and perhaps our expected value calculation assumes that this distribution is uniform, which it cannot be ... When is the image of a linear operator (between Banach spaces) closed? Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces.  When is $\operatorname{Ran}(T)$ a closed subspace?  What theorems are there?   Thanks :) In ideal gas model,  is the measure of average kinetic energy of the gas molecules. If by some means the gas particles are accelerated to a very high speed in one direction, KE certainly increased, can we say the gas becomes hotter? Do we need to distinguish the random vibration KE and KE in one direction?  Furthermore, if we accelerate a block of metal with ultrasonic vibrator so that the metal is vibrating in very high speed with cyclic motion, can we say the metal is hot when it is moving but suddenly become much cooler when the vibration stop?