$A^\mathrm{o}$, the interior of $A$, is the union of all open subsets of $A$.  If $H=\Bbb Q$ and $K=\Bbb R \backslash \Bbb Q$ then $H^\mathrm{o}=K^\mathrm{o}=\emptyset$ but $(H\cup K)^\mathrm{o}=\Bbb R^\mathrm{o}=\Bbb R$  I don't understand why $H^\mathrm{o}=K^\mathrm{o}=\emptyset$  Could somebody explain why this is the case?
I don't understand why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty, since, for every ball with the center being a rational number, given an $\epsilon&gt;0$, I can find an infinite sequence of rational numbers that approach this point. For example, take $\frac{1}{2}$. The sequence $\frac{1}{2}+\frac{1}{n}$ can be made as close as I want to the number $\frac{1}{2}$, therefore I can always have open balls with center $\frac{1}{2}$ such that there are rationals inside it.
Get the generated SQL statement from a SqlCommand object? I have the following code:  Using cmd As SqlCommand = Connection.CreateCommand     cmd.CommandText = "UPDATE someTable SET Value = @Value"     cmd.CommandText &amp;= " WHERE Id = @Id"     cmd.Parameters.AddWithValue("@Id", 1234)     cmd.Parameters.AddWithValue("@Value", "myValue")     cmd.ExecuteNonQuery End Using   I wonder if there is any way to get the final SQL statment as a String, which should look like this:  UPDATE someTable SET Value = "myValue" WHERE Id = 1234   If anyone wonders why I would do this:   for logging (failed) statements for having the possibility to copy &amp; paste it to the Enterprise Manager for testing purposes build argument lists containing whitespace In bash one can escape arguments that contain whitespace.  foo "a string"   This also works for arguments to a command or function:  bar() {     foo "$@" }  bar "a string"   So far so good, but what if I want to manipulate the arguments before calling foo?  This does not work:  bar() {     for arg in "$@"     do         args="$args \"prefix $arg\""     done      # Everything looks good ...     echo $args      # ... but it isn't.     foo $args      # foo "$args" would just be silly }  bar a b c   So how do you build argument lists when the arguments contain whitespace? How can the Anomalous Expansion of Water from 4$^\circ$C to 0$^\circ$C be explained with reference to subatomic particles? Changing the language in Windows 8.1 single language I followed the instructions in this  question  but now windows won't start properly, the screen is on but it's just black and I can see the cursor if I move it but I can't do anything else. Sum of a Hyper-geometric series. (NBHM 2011) How to find the sum of the following series $$\frac{1}{5} - \frac{1\cdot 4}{5\cdot 10} + \frac{1\cdot 4\cdot 7}{5\cdot 10\cdot 15} - \dots\,.?$$  I have no idea. I have written the general term and tested its convergence by Gauss' test for convergence, but they are neither the question nor the answer. How do I make my document use the Times font, both for the text and the math? I know that I can put \usepackage{times} in the preamble in order to use the Times font instead of Computer Modern for my document.  But this only affects the text; the math is still typeset using Computer Modern.  How can I make Latex use Times also for the math? In contemporary American English usage, I come across sentences like:   I ain’t got no money.  Ain’t no man like him.    Saying ain’t no sounds incorrect to me because it is a double negative.  What is the origin of this particular construction? I wasn’t able to find anything online related to the origins of this incorrect usage. In Bash, when to alias, when to script, and when to write a function? It's taken me almost 10 years of Linux usage to ask this question. It was all trial and error and random late-night internet surfing.  But people shouldn't need 10 years for this. If I were just starting out with Linux, I'd want to know: When to alias, when to script, and when to write a function?  Where aliases are concerned, I use aliases for very simple operations that don't take arguments.   alias houston='cd /home/username/.scripts/'   That seems obvious. But some people do this:  alias command="bash bashscriptname"   (and add it to the .bashrc file)   Is there a good reason to do that? I'm trying really hard, but I genuinely can't think of any circumstances in which I'd want to do that. So, if there is an edge case where that would make a difference, please answer below.  Because that's where I would just put something in my PATH and chmod +x it, which is another thing that came after years of Linux trial-and-error.  Which brings me to the next topic. For instance, I added a hidden folder (.scripts/) in the home directory to my PATH by just adding a line to my .bashrc (PATH=$PATH:/home/username/.scripts/), so anything executable in there automagically autocompletes.  If I needed to.  I don't really need that, though, do I? I would only use that for languages that are not the shell, like Python.  If it's the shell, I can just write a function inside the very same .bashrc:  funcname () {   somecommand -someARGS "$@" }   As I stated, I found a lot of this out through trial and error. And I only truly saw the beauty of functions when my computer died and I was forced to use the computers of the people around me when they weren't using them.   Instead of moving a whole directory of scripts from computer to computer, I ended up just replacing everyone else's .bashrc with my own, since they had never even made a single modification.  But did I miss anything?   So, what would you tell a beginning Linux user about when to alias, when to script, and when to write a function?   If it's not obvious, I'm assuming the people who answer this will make use of all three options. If you only use aliases, or only use scripts, or only use functions—or if you only use aliases and scripts or aliases and functions or scripts and functions—this question isn't really aimed at you. I need another LP-E5 type battery for my Canon 450D. Should I buy an original Canon battery or one of the many generic brands available? There are loads of cheapos on , for example.  I should add that the new battery will be added to my existing Canon battery in a battery grip.  Does anyone have any experience (good or bad) of using generic batteries? Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$? I looked at the graph of $f(x,y)=(xy)^{2/3}$ at the origin, and no matter how far I zoom in it has four corners along $x=\pm y$, so it doesn't look like it is differentiable as $z=0$ doesn't look like a tangent plane.  I also found that its partial derivatives are not continuous at $(0,0)$, e.g.  $$ \lim_{(ky^2,y) \to (0,0)} \frac{\partial f}{\partial x} = \lim_{(ky^2,y) \to (0,0)} \frac{2}{3}x^{-\frac{1}{3}}y^{\frac{2}{3}} = \frac{2}{3}k^{-\frac{1}{3}} $$ (although $\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$)  I also tried to prove/disprove its differentiability by definition, and for it to be differentiable I need $$ \lim_{(x,y) \to (0,0)} \frac{f(x,y)-0}{\sqrt{x^2+y^2}} =0 $$ but then I got stuck. Am I even on the right track?  [UPDATE] I need to do the same thing for $f(x,y)=(xy)^{\frac{1}{3}}$... I think the answer is that $(xy)^{2/3}$ is differentiable but $(xy)^{1/3}$ is not, according to my professor but I'm not sure TL2018 (MacTeX) and TikZ calligraphy library Determine the probability densities for coefficients of equation I'm not sure how to solve this problem. Can you guys provide some input/hints?      Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$.      Show that the area of $\Delta ABC$ is given by   $\frac{1}{2}\left| \det(M) \right|$, where   $$M = \begin{pmatrix} 1 &amp; 1  &amp; 1\\  x_1 &amp; x_2 &amp; x_3\\  y_1 &amp; y_2 &amp; y_3 \end{pmatrix}$$ Difference between (in / for / ∅ / during / over) the (past / last) three years Microsoft has steadily improved its operating cash flows in the past three years. Microsoft has steadily improved its operating cash flows in the last three years. Microsoft has steadily improved its operating cash flows for the past three years. Microsoft has steadily improved its operating cash flows for the last three years. Microsoft has steadily improved its operating cash flows for the past three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows for the last three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows during the past three years. Microsoft has steadily improved its operating cash flows during the last three years. Microsoft has steadily improved its operating cash flows over the past three years. Microsoft has steadily improved its operating cash flows over the last three years.  Which of them are idiomatic and preferred? Are they interchangeable? Can one create a video from sequence of meshes that you already have using blender? If I have a sequence of .obj (or any other readable format) files created from my own simulation code (say cloth simulation), can I use the rendering capabilities of blender to generate a video?   Is there a way to set the material properties for frame 1 and use the same or rest of the frames while the corresponding meshes are imported afresh from that ones I provide? Test the convergence of the series For $a&gt;0$, test the convergence of the infinite series: $$\sum a^{\ln(n)}$$ Which test should be better to use here? I tried the D'Alembert's ratio test. $$\lim_{n\to \infty} {\frac{a^{\ln(n)}}{a^{\ln(n+1)}}}= \lim_{n\to \infty}a^{ln{\frac{n}{n+1}}}=a^0=1$$ So the test failed.  UPDATE: I observed that $${a^{\ln(n)}}=e^{{\ln(a)}{\ln(n)}}={n^{\ln(a)}}={{1\over n^{ln({1\over a})}} }$$Further by test of auxiliary series $$\sum \frac{1}{n^p}$$ we know that criterion for convergence is $p&gt;1$. Here $p=ln({1\over a})$.Hence the series will be convergent when $ln(a)&lt;-1$ i.e. when $0&lt;a&lt;1/e$, since $a&gt;0$ is already given.Finally this solved my question. Why does python 2.5.2 have the following behavior  &gt;&gt;&gt;[2].extend([]) == [2] False  &gt;&gt;&gt; [2].extend([]) == None True  $ python --version Python 2.5.2   I assume I'm not understanding something here, but intuitively I'd think that [2].extend([]) should yield [2] What is the most elegant proof of the Pythagorean theorem? The  is one of the most popular to prove by mathematicians, and there are  (including one from ).    What's the most elegant proof?  My favorite is this graphical one:    According to cut-the-knot:     Loomis (pp. 49-50) mentions that the   proof "was devised by Maurice Laisnez,   a high school boy, in the   Junior-Senior High School of South   Bend, Ind., and sent to me, May 16,   1939, by his class teacher, Wilson   Thornton."      The proof has been published by Rufus   Isaac in Mathematics Magazine, Vol. 48   (1975), p. 198. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.   I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to write out all the finite subgroups in a uniform expression.  Any suggestions? Thanks in advance. What's the difference between dot syntax and square bracket syntax?