Does having a countable subbasis also satisfy the second axiom of countability?
If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
In my , I provided the following, problematic,  wording (especially bold italic), and I need help to better understand this issue so I can fix my answer:1          The thing is the books. (Reduced form of Sentence 2.a)         Fundamentally, Sentence 2.a (the so-called "correct" answer), is grammatically defective. Recall there is another grammatical rule: the subject and subject's complement should match in number.    A clarification is needed here. I am not suggesting here, in this EL&amp;U question, that there is any such singular grammatical rule as indicated in the quote above.  The quote continues as follows:     The reduced sentence makes the disagreement between the subject's and complement's plurality obvious. What is one to do? Language is linear and we know "The thing is X" is better than "The thing are X", so we go with the former, which is the subject-verb agreement rule. But what is "the books" then? It must be thought of as a collective noun, even if it doesn't look or feel like one. Forcing something to be a collective noun is related to the idea of notional agreement.   Others noted that grammatically defective is too strongly worded.  There is no such "grammatical rule" for subject/complement agreement.  Please help me understand/improve upon my line of thought here.  I'm stating that Sentence 2.1 presents a fundamental "conflict" or "error" (however one might define error) at some level (morphological, syntax, grammatical, semantic):     The thing is the books.  (Or, "The thing is the objects.")   Based on the "obvious" morphosyntactic plurality conflict, the above sentence feels so wrong that I think many native speakers would think it "simply" grammatically incorrect. However, @F.E. gives counter examples:     And here are some more grammatical examples, where the number of subject and predicative complement don't agree: "They were a problem to us all", "That so-called work of art is simply four pieces of driftwood glued together" (CGEL, page 254-5). As CGEL says: "What is required is semantic compatibility, not syntactic agreement . . .". There are more examples on page 512: "The only thing we need now is some new curtains", "The major asset of the team is its world-class opening bowlers", "Our neighbors are a nuisance", "This gadget is five different tools in one".   I think all the examples given above can be explained as notional agreement.    Question 1: If we explain this phenomena in terms of notional vs syntactic agreement, where does "grammatically correct/incorrect" fit in?  Question 2: Can a simple sentence such as "The thing is the objects." be grammatically incorrect while more complex sentences such as "The thing is four pieces of driftwood glued together." be grammatically correct?  (The answer, in my mind, must be no; this is my conundrum.)  Question 3: Would it be better to say that Sentence 2.a has a low level of (linguistic) ?   Question 4: How about a low level of  and that it's semantically difficult to be (linguistically) ?    Question 5: Do we simply draw a hard line and say "The thing is the objects." is 100% grammatically correct?  I didn't particularly like this option based on intuitive notions of "grammatically correct".    1. Please help me and be kind; I'm trying to improve my understanding. I'm looking for cool, objective advice on how to look at this.  I numbered my questions so they can be easily referenced, if desired.  My preference is the final answer would sufficiently answer all the questions, but of course that can be done explicitly or implicitly. Insert results of a stored procedure into a temporary table Use the new helpcenter for the question/answer ban message Related:    The helpcenter has . It still links to the  for further details. Besides that, the question ban message .  Now that we have the helpcenter, could the following changes be implemented?   Link to the help center instead of the meta post Remove the link to the meta post from the help center, and instead incorporate its contents into the helpcenter itself I have a Windows 10 Home PC that is often unattended but doing important work. The work follows no particular schedule, and may take place at any time of day or night.  As things stand, Windows 10 (anniversary update) is configured to automatically restart and install updates during inactive times. The user can configure the inactive times, but the OS forces the user to have no more than 12 active hours a day. This means that the machine may well choose to restart at a time that is highly disruptive to our work and when there is no user around to prevent the restart.  For this reason, I would like to ensure that Windows never automatically restarts. How can I achieve this? What's better to use in PHP for appending an array member,  $array[] = $value;   or  array_push($array, $value);   ?  Though the manual says you're better off to avoid a function call, I've also read $array[] is much slower than array_push(). What are some clarifications or benchmarks? Having problems with hair transparency using cycles. I ported over a model from daz studio (have done this a few times before with no issues).   Using a simple mix shader, transparent top, diffuse bottom, black/white texture in the fac.  Heres the blend  with images packed. There has been some discussion about getting notified on comments to questions or answers that don't belong to you, and the long-standing wish to be able to keep track of new answers or developments in questions asked by other people.  Previous discussion in regards to subscribing to questions:                  (very similar to this one)      In regards to subscribing to comments:                    The general feeling - at least here on meta - seems to be against broadening the notification rules, out of fears of useless notifications to flood users' inboxes, turning SO into more of a discussion forum than it should be, etc.   Here's a suggestion that would keep the user 100% in control what they get notified about, not generate any notifications by default, and require only minimal modifications to the User Interface.  Each question and each comment area would get an envelope symbol.    it is inactive (=white) by default.  When those envelopes are hovered over, the tooltip says click to subscribe or something.  When clicked on, the envelope lights up in green or some other colour (but different from the actual notification envelope's orange to avoid confusion.) You are then subscribed to any updates to that question or comment thread.    For each subscription, you receive notifications in the "responses" tab, as if you were the owner of the comment or question.     To unsubscribe from a question or comment thread, you open the page, and click the green envelope. It turns back white, and the notifications disappear from your responses tab (if technically possible - otherwise, there just won't be any new ones added.)    No further functionality, like managing subscriptions or anything, is necessary IMO. If the flood of notifications from a subscribed question starts to annoy you, you just open it and unclick the envelope. Done.   It would make it possible to leave a comment to a newbie's question and make sure their answer reaches you even if they don't know they need to address you using @username You could subscribe to those "oh, I'd like to know the answer to that" questions and receive active notification, instead of them getting forgotten in your favourites list You could subscribe to discussions you are interested in the outcome of, but don't want to participate in. It would (in my opinion) sort out a number of woes aired by many users in a good way, without adding annoyances to those who like the system just fine the way it is. It would not require a change in the visible layout. As far as I can see, the space the envelope would need to be is empty everywhere.   I don't know SO under the hood of course, but technically, the system would be doing what it's doing already, notify question owners of updates - just that the circle of internal "owners" would grow. Setting a custom RGB background color in beamer I believe I've correctly followed the manual in defining a custom background color in the example below.     But as the example shows, the background color is black rather than an extremely pale beige.    Could anybody advise what I'm doing wrong please?  \documentclass{beamer} \definecolor{MyBackground}{RGB}{1.0000    0.9451    0.6549} \setbeamercolor{background canvas}{bg=MyBackground} \begin{document} \begin{frame} Hullo world \end{frame} \end{document} What is the relationship between ISO, aperture, and shutter speed? How to make `sudo` preserve $PATH? I have a program that is installed in a custom directory under /opt. To make it easier to run it, I edited my bashrc to add said directory to my path: export PATH=$PATH:/opt/godi/bin:/opt/godi/sbin  This works fine if I want to run the program without sudo. However, if I try to run it with sudo it fails with a &quot;command not found&quot; error. $ sudo godi_console sudo: godi_console: command not found  Inspecting the PATH variable after using sudo reveals that its not including the same PATH I have as a normal user: $ sudo sh # echo $PATH                  /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin  Why is the PATH not the same? Am I doing something wrong? I'm on Debian Jessie, if it makes a difference. One thing I tried was to invoke /opt/godi/sbin/godi_console directly, passing the absolute path to the executable. Unfortunatelly, that didn't help in this particular case because godi_console itself depends on the PATH being correctly set. Prove $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$   So, LS= $$\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}$$ $$\left(\dfrac{1}{\sin\theta}\cdot \dfrac{\tan\theta}{1}\right)-\left(\dfrac{1}{\tan\theta}\cdot \dfrac{\sin\theta}{1}\right)$$ $$\dfrac{\tan\theta}{\sin\theta}-\dfrac{\sin\theta}{\tan\theta}$$ Now, considering the fact that I must have a common denominator to subtract, would this be correct: $$\dfrac{\tan^2\theta}{\tan\theta\sin\theta}-\dfrac{\sin^2\theta}{\tan\theta\sin\theta}\Rightarrow \dfrac{\tan^2\theta-\sin^2\theta}{\tan\theta\sin\theta}$$ I feel like I'm close to the answer because the denominator is the RS of the OP. Please help. Do not give me the answer. If $x+{1\over x} = r $ then what is $x^3+{1\over x^3}$? If $$x+{1\over x} = r $$ then what is $$x^3+{1\over x^3}$$  Options: $(a) 3,$ $(b) 3r,$ $(c)r,$ $(d) 0$ Solutions for INSERT OR UPDATE on SQL Server Assume a table structure of MyTable(KEY, datafield1, datafield2...).  Often I want to either update an existing record, or insert a new record if it doesn't exist.  Essentially:  IF (key exists)   run update command ELSE   run insert command   What's the best performing way to write this? How can I play sound in Java? What should I do if my Windows computer seems to be infected with a virus or malware?  What are the symptoms of an infection? What should I do after noticing an infection? What can I do to get rid of it? how to prevent from infection by malware?   This question comes up frequently, and the suggested solutions are usually the same. This community wiki is an attempt to serve as the definitive, most comprehensive answer possible. Feel free to add your contributions via edits. queries with db_select don't respect integers When I check "SHOW PROCESSLIST" in phpMyAdmin I can see lots of queries where argument-integers are being inserted as strings (with single quotation marks).  I'm wondering if this is due to the way phpmyadmin reports the queries (if it puts every argument between quotation marks) of if it is db_select that puts every argument between quotation marks.  An example of my code where integers are being used as arguments:  $select-&gt;join('taxonomy_index', 't_'.$key, 't_'.$key.'.nid = n.nid AND t_'.$key.'.tid = :tid_' . $key, array(':tid_'.$key =&gt; (int)$tid)); $select-&gt;condition('t_'.$key.'.tid', (int)$tid);   As you can see I explicitly write (int)$tid but in my query log I can see the following:  ... WHERE (t_0.tid = '10') ...   The big question is: How can I be sure that integers are being treated as integers in my queries? I don't want MySQL having to transform strings to integers. no space left in '/' but a lot space left in '/home' i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable. I have been given the example $$ f_n(x)= \begin{cases} 1, &amp; \text{if}\; x\in\{(x_k)_{k=1}^{n}\}, \\ 0, &amp; \text{otherwise.} \end{cases} $$ Here the sequence $(x_k)_{k=1}^{\infty}$ is the sequence that "enumerates" elements of $\mathbb{Q}$. The sequence of functions converges pointwise to $$ f(x)= \begin{cases} 1, &amp; x\in\mathbb{Q}, \\ 0, &amp; x\notin\mathbb{Q}, \end{cases} $$ which isn't Riemann integrable. However, I am not quite convinced that $f_n$ is integrable. I would like some help in seeing how $f_n$ is Riemann integrable. Also, I would like more examples of sequences of Riemann integrable functions $f_n$  that converge pointwise to a function $f$ that is not Riemann integrable. How do I get Skyrim to recognize I don't have a follower anymore? Probability of flipping coin What is the probability of obtaining THT before HHH when a toss is tossed sequentially? Could someone solve using state diagram and law of total probability?