If $A$ is a connected set and $\{A_i : i \in I\}$, $I$ an arbitrary set (can be countable or not) of connected sets.  How to show that if $A \cap A_i \neq \emptyset$ for all $i \in I$ then $A \cup (\cup_{i\in I} A_i)$ is connected?  I am trying to show that if $A \cap A_u \neq \emptyset~~ \forall i \in I$ then for all $i, j \in I$ $A_i\cap A_j \neq \emptyset.$ This enought to conclude the result. But I stuck here.  What I tried: Suppose that there are $i,j\in I$ such that $A_i \cap A_j  = \emptyset.$  Then since $A\cap A_i \neq \emptyset$ and $A\cap A_j \neq \emptyset$ it somehow induces me to think that is possible to obtain a split for $A$. I don't know how to proceed.  Thanks
Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected.   If $\bigcup\mathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $\bigcap\mathscr{F}$. Then either $x\in A$ or $x\in B$. I don't know where to go from here.
Write a formula in an Excel Cell using VBA I'm trying to use VBA to write a formula into a cell in Excel. My problem is that when I use a semicolon (;) in my formula, I get an error:     Run-time error 1004   My macro is the following :   Sub Jours_ouvres()     Dim Feuille_Document As String     Feuille_Document = "DOCUMENT"             Application.Worksheets(Feuille_Document).Range("F2").Formula = "=SUM(D2;E2)"     End Sub How to insert text before the first line of a file? I haven't written any C++ in years and now I'm trying to get back into it. I then ran across this and thought about giving up:  typedef enum TokenType {     blah1   = 0x00000000,     blah2   = 0X01000000,     blah3   = 0X02000000 } TokenType;   What is this? Why is the typedef keyword used here? Why does the name TokenType appear twice in this declaration? How are the semantics different from this:  enum TokenType {     blah1 = 0x00000000,     blah2=0x01000000,     blah3=0x02000000 }; I understand that if a particle approaches a finite potential barrier of height $V_0$ with energy $E &lt; V_0$, there is still a finite probability of finding the particle on the other side of the barrier due to quantum tunneling.  My question is, since the wavefunction is nonzero inside the barrier region, is it possible to actually make a position measurement and locate a particle inside the barrier?  I mean, if we can say that "there is a nonzero probability that the particle is inside the barrier", surely this would suggest we can do so?  If not, why not? Am I understanding the whole wavefunction/probability distribution thing right? How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? Proving the set of points at which a function diverges to $\infty$ is countable Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set    $$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable.   My book told me to consider  $g(x)=\arctan(f(x))$, then it said "it is easy to see the set is countable." But I still can't understand what it mean. I recently upgraded Windows 7 to Windows 10. The biggest issue I have now is that the Windows Search service no longer works, making the Start menu mostly unusable.  If I go to Indexing Options in the control panel I see that Indexing is not running and the list of paths to index is empty.  Going to advanced options, the current index path is empty, and the new path is empty too. This is true even if I try to use the "New Index" button.  Using the troubleshooter won't help since the service doesn't start. If I try to manually start the "Windows Search" service, I get the error  "Error 3: Unable to find specified path" (this is the rough translation, the original message is in my language).  What can I do to find out which path/file is missing ? I tried to check with Process Monitor, but the only file (beside system libraries) the SearchIndexer.exe try to access seem to be c:\ProgramData\Microsoft\Search\Data\Application (everytime the service is started the whole folder c:\ProgramData\Microsoft\Search is deleted and recreated) and C:\Windows\System32\config\systemprofile (which exists).  Edit: all available updates are installed, and issuing      SFC /scannow   Doesn't reveal any issue Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$. Let $K$ be the rational function field $k(x)$ over a perfect field $k$ of   characteristic $p &gt; 0$. Let $F = k(u)$ for some $u\in K$, and write   $u = f(x)/g(x)$ with $f$ and $g$ relatively prime. Show that $K/F$ is   a separable extension if and only if $u\notin K^p$.   There is already a post on this but it has not yet been answered :   The truth is that I am very confused with this problem, I would like someone please help me to understand what happens with this exercise and explain to me how it could be done by giving me a help or something. I already have an approach for the left-to-right implication, but I do not know very well how good it is. Could someone help me with the other implication please? Thank you very much. Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$ How exactly do automatic updates work? I received an email this morning stating that my Wordpress site had been automatically updated to the latest version. I knew about the feature but I've always wondered exactly how it works.  PHP isn't a permanently-running process: it only runs when requested. So as far as I can tell, Wordpress can only update itself when someone loads a web page. But the update process is not instantaneous, so surely a user visiting the site would have a really slow page load.  Is there a different trick they use for automatic updates? I've searched all over the place but haven't found any explanation. How full is too full for mechanical hard drives? I have heard many claim that it doesn't matter how full a drive is until it starts cutting into temp and virtual memory space.  This doesn't make sense to me, given the nature of how the data is transacted on a hard drive. The inside of the platter presents less data per revolution than the outside of the drive does, by significant factors. The inside 40% of the radius of full size hard drive is used for the spindle, so only the outside 60% is used for data storage, but that still means that the inside track of a hard drive presents data 60% slower than the outside track.  By my calculation, a Hard drive that is only 10% full should perform about 2.25 times faster than a hard drive that is 90% full, assuming that the flow is constrained by other factors.  Am I wildly off base here? For all the drives I know, even the top speeds of the first 1% of the drive would be well within the bandwidth provided by a SATA 2 connection. How can I turn on the Intel HD manually? For some weeks my MacBook 6.2 does not switch to the integrated graphics adapter (Intel HD). This is especially crucial when I work on battery (which happens quite often). I guess the runtime could be enhanced when it would take the HD adapter for display.  I checked that it automatically changes the graphics mode. I am running Snow Leopard 10.6.8.  I recently installed 8GiB RAM (not from Apple). Could this probably related to this issue? Actually I do not want to downgrade or use the expensive Apple RAM.  So, how can I manually turn on the internal graphics adapter? Change the URL in the browser without loading the new page using JavaScript How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? I’m looking for an alien movie I watched years ago, mid to late 2000’s about the Earth being cooked or heated up by aliens. The poster is a reflection of the Earth heating up with an orange glow on an aliens eye. It was a low budget movie and I think the one I watched was pirated from China. Again low budget, bad quality and the name could’ve been something worthy he word scorch or something like that. The alien in the movie had legs with knees that went inwards and it had red eyes. Number formation from given digits I was asked this question in a recent interview. You are given two digits $0$ and $9$. Using only these two digits you must form the least positive number that is a multiple of a given '$n$' . The value of $n$ will be provided before hand. For example if we are given $n$ as $7$ then the answer will be $9009$ .  I tried using some basic divisibility principles but was not able to come up with a correct solution and code it up. Probably I need some help in number theory This has been a old live-action movie from Asia, most likely Japan. I've seen it some time in the late 80's and I think it's been black/white only.  Earth is invaded by UFOs (small and round, like flying balls), coming from Mars.  Japan (I think) built some kind of super spaceship, initially hidden within some secret port in some kind of island/mountain.  The spaceship looks similar to a classic battleship, but with an added drill below the front.  In the movie's finale it's revealed that this is some kind of   super weapon/bomb that has to be manually piloted by someone. When launched it starts rotating and flies (formed like a bullet but with the drill/corkscrew head).   To defend Earth, the ship is sent on a mission to Venus, where the UFOs seem to originate. I don't remember the specific reason or conclusion here.  The ship starts underwater or launches from the mountain on the island, then taking off and fighting UFOs while leaving Earth's atmosphere. Why are two different concepts both called "heap"? Can I change the color of the last selected face, so that it is the same as the other selected faces?  Here, I've selected the left face then the right face. Note the right face is some sort of checkered or dithered pattern, and just the outline has the nice orange 'selected' color.    Here I've selected the top face, note the right face has now the nice color.    And here I've deselected the top face. Now, both of the remaining selected faces have the nice color. This is the result I'd rather see instead of image 1. Note that this is not the same as:   Mistagged questions should be retaggable immediately.