I wanted To show that if E is connected subset then $\bar E$ is also Connected By Using Definition. There is one answer regarding this question but that does not use Definition So this question is not repeated .My attempt: On contrary $\bar E$ is disconnected so  $\bar E$=$A \cup B$ Where A and B are disjoint open set .I had only info about E is connected that means E can not be written like above . From Hint:E$\subset$ $\bar E$=$A \cup B$ So E=$(A\cap E) \cup (B\cap E)$   $(A\cap E)$ and   $(B\cap E)$ are both open and disjoint Because In case x$\in(A\cap E) \cap (B\cap E)$ Then x$\in A \cap B$ Which contradicts with Given .Is this is enough to show ?
I have to prove that if $E$ is a connected space, then so is $\overline{E}$ (the closure of $E$) a connected space. I tried to prove the contrapositive. So suppose that $\overline{E}$ is not connected in a metric space $X$. This implies there are $A\subset X$ and $B\subset X$ such that $A\cap B=\emptyset$, $A\cap \overline{E}\neq \emptyset$, $B\cap \overline{E}\neq \emptyset$ and $\overline{E}\subset A\cup B$. I will try to prove that this $A$ and $B$ will also work for $E$. Suppose that $A\cap E=\emptyset$ and $B\cap E=\emptyset$, this implies that $A\cup B$ is a subset of the limit points of $E$, because $A\cap \overline{E}\neq \emptyset$ and $B\cap \overline{E}\neq \emptyset$. This contradicts the fact that $\overline{E}\subset A\cup B$, so we have to conclude that $A\cap E\neq \emptyset\neq B\cap E$. Since $E\subset\overline{E}\subset A\cup B$, we have that $E$ is also not connected.  My question is whether my proof is correct, because I have a little bit doubt. If it's not correct, what is the best I can do? Thanks in advance!   Edit1: The definition we use for a connected space $E$ in a metric space $X$ is that if $E$ is connected, then there are no open $A,B\subset X$ such that $A\cap B=\emptyset$, $A\cap E\neq \emptyset\neq B\cap E$ and $E\subset A\cup B$.  Edit2: $A$ and $B$ must be open.
Can you use an alias in the WHERE clause in mysql? I need to use an alias in the WHERE clause, but It keeps telling me that its an unknown column. Is there any way to get around this issue? I need to select records that have a rating higher than x. Rating is calculated as the following alias:  sum(reviews.rev_rating)/count(reviews.rev_id) as avg_rating How to fix apt: Signature by key uses weak digest algorithm (SHA1)? Why can't I find a USB-C hub with multiple USB-C ports? My computer only has one USB-C port and I want to connect multiple USB-C devices to it.  I have trawled through Amazon but am amazed that I cannot find a single one. There are many hubs that have USB 3.1 USB-A ports but not USB-C ones.  Is there a limitation in the spec that prevents this being possible? The first day of the current month in php using date_modify as DateTime object poll() failed, context incorrect? - Example: bpy.ops.view3d.background_image_add() I keep on getting context errors in the Blender Python API.  I don't understand the issue of the context in the API, how do I know which is the correct context?   Thanks! Does $A$ a UFD imply that  $A[T]$ is also a UFD? I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD.   The only thing I am sure I could try to use is Gauss's lemma.  Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\ldots,x_n]$ and $k[x_1,\ldots,x_n]$ are UFDs? Is there a setup where I can leave the water on all summer? public class doublePrecision {     public static void main(String[] args) {          double total = 0;         total += 5.6;         total += 5.8;         System.out.println(total);     } }   The above code prints:  11.399999999999   How would I get this to just print (or be able to use it as) 11.4? MySQL Syntax error message "Operand should contain 1 column(s)" I tried running the following statement:  INSERT INTO VOUCHER (VOUCHER_NUMBER, BOOK_ID, DENOMINATION) SELECT (a.number, b.ID, b.DENOMINATION)  FROM temp_cheques a, BOOK b WHERE a.number BETWEEN b.START_NUMBER AND b.START_NUMBER+b.UNITS-1;   which, as I understand it, should insert into VOUCHER each record from temp_cheques with the ID and DENOMINATION fields corresponding to entries in the BOOK table (temp_cheques comes from a database backup, which I'm trying to recreate in a different format). However, when I run it, I get an error:  Error: Operand should contain 1 column(s) SQLState:  21000 ErrorCode: 1241   I'm running this in SQuirrel and have not had issues with any other queries. Is there something wrong with the syntax of my query?  EDIT:  The structure of BOOK is:  ID  int(11) START_NUMBER    int(11) UNITS   int(11) DENOMINATION    double(5,2)   The structure of temp_cheques is:  ID  int(11) number  varchar(20) This is intended to be a canonical question regarding possible endings of the game.  For each question, assume a single proper strike was made, and note where an improper strike would change the game outcome (beyond an additional point for the opponent).  If the Queen has been covered:     What happens if I pocket my last C/m and the Striker? What happens if I pocket my opponent's last C/m? What happens if I pocket my opponent's last C/m and the Striker? What happens if I pocket both players' last C/m? What happens if I pocket both players' last C/m and the Striker?     If the shot covers the Queen:     What happens if I pocket my last C/m and the Queen? What happens if I pocket my last C/m, the Queen and the Striker? What happens if I pocket my opponents' last C/m and the Queen? What happens if I pocket my opponents' last C/m, the Queen and the Striker? What happens if I pocket both players' last C/m and the Queen? What happens if I pocket both players' last C/m, the Queen and the Striker?     If the Queen is left on the board:     What happens if I pocket my last C/m, leaving the Queen? What happens if I pocket my last C/m and the Striker, leaving the Queen? What happens if I pocket my opponent's last C/m, leaving the Queen? What happens if I pocket my opponent's last C/m and the Striker, leaving the Queen? What happens if I pocket both players' last C/m, leaving the Queen? What happens if I pocket both players' last C/m and the Striker, leaving the Queen? How should one maintain and update a MiKTeX installation? Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? Show bounded and convex function on $\mathbb R$ is constant I have two different machines (home and work) running Ubuntu 18.04. Last night vim froze at home. I was in insert mode and typing and went to save (esc :w) and nothing happened. The status bar still reads -- INSERT --, the cursor is still blinking where it was. I was stuck. I couldn't find a way out. I couldn't type (nothing happened when I type), I couldn't move around (the up and down arrows did nothing). It was stuck in insert mode with the cursor blinking where it was.   I was definitely multitasking and probably hit some other keys in there, but I don't know what keys. It was late, though, so I closed the terminal window and tried again (I was entering a git commit message). It happened again partway through my typing so I switched to git commit -m "don't need an editor for this" instead. And then I shut down my computer and stopped working.  I figured I was just tired, but then it happened to me today at work on a different laptop altogether. Again I was multitasking and can't swear I didn't type any bizarro key sequence but if I did it was accidental. And other tabs in the same terminal aren't frozen.   I'm used to getting trapped in visual mode in vim. That's a trick I've learned. But stuck in insert mode? Any ideas on what I might've done and how to get out of it?   Per a comment suggestion I tried looking at .viminfo but the only .viminfo I see is owned exclusively by root and only appears to show things I would have edited with sudo:   # Input Line History (newest to oldest):  # Debug Line History (newest to oldest):  # Registers:  # File marks: '0  1  0  /etc/neomuttrc |4,48,1,0,1531789956,"/etc/neomuttrc" '1  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,49,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" '2  51  0  /etc/apt/sources.list |4,50,51,0,1530816531,"/etc/apt/sources.list"  # Jumplist (newest first): -'  1  0  /etc/neomuttrc |4,39,1,0,1531789956,"/etc/neomuttrc" -'  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,39,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" -'  1  66  /etc/apt/sources.list.d/signal-bionic.list |4,39,1,66,1530816565,"/etc/apt/sources.list.d/signal-bionic.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  51  0  /etc/apt/sources.list |4,39,51,0,1530816531,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list" -'  1  0  /etc/apt/sources.list |4,39,1,0,1530816447,"/etc/apt/sources.list"  # History of marks within files (newest to oldest):  &gt; /etc/neomuttrc     *   1531789952  0     "   1   0  &gt; /etc/apt/sources.list.d/signal-bionic.list     *   1530816564  0     "   1   66     ^   1   67     .   1   66     +   1   66  &gt; /etc/apt/sources.list     *   1530816454  0     "   51  0   It seems odd that I wouldn't have an unprivileged .viminfo but I did sudo udpatedb and locate .viminfo and still didn't surface more than the one root-owned file. Cat5e cable has a rated current of 0.577A (). I am wondering if this is voltage dependent? I am almost certain it is not, but if it is, something might blow up. Say I need to get 2A at 5V across the cable. I boost it to 24V / 0.5A, send it across the wire, and step it down to back to 5V and a bit under 2A. Would that work? If the cable is proper Cat5e could I do this trick with up to 100V as long as I don't exceed 0.577A?   I am worried if the cable heats up at near maximum rated current, for instance, the safe voltage could drop as the insulation softens up or something. I will certainly test this, but I wonder if there is some rule that I don't know about. Can I install Android on my non-Android device? Many people want to try/use Android without buying a new Android phone.  How can I install Android on my current phone or device running Nokia Symbian, Maemo, or MeeGo; Microsoft Windows Mobile or Windows Phone 7/8/10; Apple iOS (iPhone); RIM's BlackBerry OS; Palm's WebOS; Samsung's Bada; HP's WebOS; Mozilla's Firefox OS; or other phone / operating system? Are there any official specifications regarding the torque for camera mounts? I've been wondering why I've never seen any official specifications of the maximum allowed torque that can be exerted upon lens mounts. It seems to me like a very useful specification, but I can't seem to find it from any manufacturer.  indicates that there might be no specifications for Nikon mounts. Are there really no manufacturer that publish these specification of their mounts? How does the linux kernel know device major and minor numbers? As far as I know, the kernel detects hardware, adds information to sysfs creates a device in /dev and then generates a udev event. My question is, do device drivers do all of this or it is the kernel itself? If drivers do it, then they would know the device major and minor number to create the file in devtmps.   I also understand (maybe wrongly) that it is udev who calls modprobe to load the drivers that handle a device.   I was told that they do so with the contents of the MODALIAS uevent environment variable, thus loading all modules whose aliases match this string (the correct driver). In short: what are the steps of hardware detection in linux and WHO performs each step: the kernel itself, the driver inside the kernel or udev in userspace. Thanks. How do I configure a new Ubuntu installation to accept ssh connections? Is there a subtle difference between "somebody" and "someone", "anybody" and "anyone"? According to the Page 87 in Kruschke: Doing Bayesian Data Analysis, the author says that the mean of a distribution is a value that minimizes the variance of a probability distribution, for example, a normal distribution. The following is what is mentioned in the page:  It turns out that the value of $ M $ that minimizes $ ∫dx p(x)(x−M)^2$ is $E[X] $. In other words, the mean of the distribution is the value that minimizes the expected squared deviation. In this way, the mean is a central tendency of the distribution.  I have read the paragraph and kind of understood what the author is trying to say but I wonder how this can be written mathematically using the above equation or any other ways.