Please give me examples of disconnected metric spaces that cannot be isometrically embedded into connected metric spaces.   I have just started to study connected metric spaces and I haven't found out a metric space $X$ which isn't a subspace from another metric space such that $X$ isn't connected.  Every counterexample that I've looked, has been subspaces from a bigger metric space, e.g. $[0, 1]\cup[2, 3]$ which fails to be an interval in $(\mathbb{R}, |\cdot|$).   It might be topological space that isn't connected but for metric spaces? is there any apart from discrete metric spaces?  Thanks in advance
Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected.  I'm not sure whether this will work or not, but intuitively I think the answer is yes.
With powershell get exactly the same application list as in Add/Remove programms I am using two books for my calculus refresher.   Thomas' Calculus  Higher Math for Beginners by Ya. B. Zeldovich   My question is : When applying Integral Calculus for calculation of volumes of solids, generated by curves revolved around an axis, we use slices of 'cylinders' to approximate the volume of the resulting solid and then integrate the sum of those infinitesimal cylinders. However, when we are using the same techniques to calculate the surface area of the surfaces generated by revolving curves around an axis, we consider the 'slope' of the differential length of curve 'dx', calculate the total length of the curve and derive the required surface area.  Are we not supposed to use the same 'slope' for calculating the volumes of the infinitesimal 'cylinders' for calculation of volumes? Shouldn't we use 'sliced portions of 'cones' as the infinitesimal volumes?? When it come to calculation of volumes of solids of revolution, why are we neglecting the slope of the curve for the differential length and simply assume that it is an infinitesimal cylinder??  Ex: Let us say we want to calculate the surface area and the volume of the solid generated when the parabola y = 10 . x^2 is revolved about the y-axis, with limits of x from 0 to 5.  In such cases, when calculating the volume of the solid, we consider infinitesimal 'cylinders', ignoring the 'slope' of the curve for the differential element 'dx', but when calculating the surface area, we consider the 'slope' of the differential element 'dx'. How do I rename a domain and preserve SEO? As part of graduating from Area51 and becoming a full fledged site we sometimes change domain names from http://topic.stackexchange.com to http://example.com. Aside from the obvious 301 redirect rules to support existing urls transparently, are there other techniques that would aid in the preservation of SEO? In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count. What can I use to find duplicate photos, including photos that have been resized? Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime. I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$. I then need to show this implies p isn't a gaussian prime.   I have started to solve this using Wilson's theorem that a number $z$ is prime iff $(z-1)!\equiv-1\pmod z$. Therefore the endpoint of my proof should be that $(p-1)!\equiv-1\pmod p$.  As $p$ is of the form $p=4n+1$, I only need to prove that $4n!$ is congruent to $-1$ modulo $p$.   Here is my working so far: Starting with the congruence $x^2 \equiv -1\pmod p$:    $$\eqalign{x^2 = -1\pmod p&amp;\implies   x^2 + 1 = kp \implies (x-i)(x+i) = kp\cr &amp;\implies ((2n)!-i)((2n!)+i))=kp \implies 4n!- 1 =kp\cr}$$   This is where I start to run out of any ideas that seem to get me anywhere. Any tips would be greatly appreciated! How to get dark menu bar and dock in Mojave light mode? I prefer light windows with dark menu bar as in High Sierra. However, it seems I can only choose all-dark or all-light in Mojave now. Are there any way to hybrid or customize it? Why does Sass change the format of my colors? Fantasy book with shape shifting dragon warriors Set in a fantasy world with shape shifting Dragon kings and their warriors. They terrorize the humans, and kill those with magic.   The protagonist is the great-grandson of a powerful warlock. He finds out that he has magic, and fights with it to save his new friends, the Griffin, Blackhorse (I think that's his name), some creepy necromancer who is immortal, and Gwen (fellow sorceress), along with the kingdom that the Griffin rules in.   They fight the protagonist's father, the leader of the Dragon kings, the king's Duke, and the creepy sorcerer, who has a curse (when he dies, he changes side from good to bad, or bad to good). I have no clue what this is called, but I want to find out, since I'm out of things to read. Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$ Let $x_1 = 2$, and define $$x_{n+1} = \frac{1}{2} \left(x_n + \frac{2}{x_n}\right).$$ Show that $x_n^2$ is always greater than or equal to $2$, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = \sqrt2$.   My question:  So I know how to do this problem but I don't know how to prove that $x_n &gt;0$ for all $n\in \mathbb{N}$. How to partition an encrypted disk I've installed Ubuntu 13.10 and formatted my whole HDD ,and i chose to encrypt the disk and it automatically chose the LVM option (and put it as a logical partition) now after Ubuntu installed i tried to repartition the disk thru Live CD put GPartition gave me it's an encrypted partition and it can't be resized , i want to put the system's files and the home folder on partition and my private files on another one  Any help please ? A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal.   These are exactly what we want for a physical observable. I think this is the all required.   So why is quantum mechanics not satisfied with symmetric operators, but wants more? How does negation affect the use and understanding of "or" and "and" I'm trying to make more sense of how negation effects how a sentence is parsed and understood if and's and or's are used within them.  Pop quiz: You are trapped on a bus with a bomb going 50 MPH. You have a radio... (okay enough Keanu Reeves references) :-)  Let's say the bomb expert says "Don't cut the red or green wires." I think that means "Do not cut the wires colored red. Do not cut the wires colored green."  Now, let's say the bomb expert said on the radio "Do not cut the red and green wires", instead. Do I take that to mean:     "Do not cut the wires colored red. Do not cut the wires colored green."   OR     "Do not cut the wires whose insulation has red and green stripes." ?   OR     "It's okay to cut the wires colored red or the wires colored green, as long I don't cut one of each color."   Which meaning should I take if the radio breaks and I can't ask follow up questions? SQL-Server: The backup set holds a backup of a database other than the existing BAV99 forward voltage I am reading the  From the table on ELECTRICAL CHARACTERISTICS (page 2) (TA = 25°C unless otherwise noted) (Each Diode)     Forward Voltage mVdc     IF = 1 mAdc 715mV   IF = 10 mAdc 855mV   IF = 50 mAdc 1.0V   IF = 150 mAdc 1.25V     On page 3 there is the following VI curve graph:  Clearly at 25°C - 100mA is Vf = 0.9V. There is huge difference between the table and the IV curve.  Which source of information I should trust? Is this (typical) and (maximum) where the table guarantees the diode won't exceed Vf 1.0v at 50mA while the graph is just typical values? How can number of subdirectories and files be determined from the output of using ls -ld command?  I realize that this command only lists the directories. How do electric currents make light? In an incandescent light bulb, electrons flow through wires to the bulb, where is passes through a filament, and the filament lights up.  Does this work because of several high energy electrons losing some of their energy to disturb the electromagnetic field, creating photons? The electrons just leave the filament afterward and are replaced by more high-energy electrons from the negative terminal. Or does it work differently?  As a secondary question, is there a type of light bulb that directly converts electron flow to photons? I'm a computer student, learning math just for fun. Today I was  that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to converge to 1 when there are even powers and to 0 when there are odd powers! Then I attempt to prove it but I failed.  Then I did a little research and I found the  article on Wikipedia. This article says that my guess was right but without any proof. So I'm here to ask you about it.     If we define ${x^{x^{\cdot^{\cdot^{x}}}}}$ as ${^{n}x} :=\begin{cases} 1 &amp;\text{if }n=0 \\ x^{\left(^{(n-1)}x\right)} &amp;\text{if }n&gt;0 \end{cases}$      then prove:      $$\lim_{x\rightarrow0} {}^{n}x = \begin{cases}   1, &amp; n \text{ even} \\  0, &amp; n \text{ odd} \end{cases}$$ For users running Java application or applets in the browser, what difference does it make running either Sun Java or OpenJDK?  EDIT  Oracle will be , and basing the proprietary implementation on OpenJDK. Can we expect similar experience in running Java apps using either OpenJDK or Oracle JDK? I took this photo of the blood moon this morning and I am finding many of my photos, especially landscape photos, have a significant lack of clarity. I'm not looking for a photo critique because I know this particular photo is crap. I'm looking to identify my source of failure to nail a significantly clear photo.     Setup  Canon EOS 70D with a 100-300 telephoto lens (I'm sorry I don't know the exact model. It's from Canon but not one of their professional quality lenses.)   Aperture: f/5.6 Shutter: 1 sec. ISO: 1000 Focal Length: 300mm Metering mode: Spot White Balance: I manually set the WB to 5500 kelvin, it looked fine in live view. Focus Mode: Manual Drive: 2 sec timer delay   Mounted on a decently stable tripod  Process  I turn on live view, and square up the target in the display. I then set my focal length, focus in using live view, adjust the focus as sharp as possible and take the pic. I've noticed that in this case, the picture was incredibly grainy, which doesn't translate to the final photo, but wonder if it has anything to do with it.  This particular shoot I found it difficult to balance out the variables with a dark sky and a moving moon that starts blurring after 1-2 seconds of exposure, which is why I had the aperture wide open and the ISO cranked up.  I can't tell where I'm failing:  Equipment:  The 70d takes great photos but the telephoto is not great when it's totally zoomed in. My wallet doesn't hold enough cash for a motorized mount to track objects in the sky.  Circumstance: I've seen great photos of the moon with the use of a camera and a tripod, so I know this is possible. Although, I keep seeing pictures of this big gorgeous moon over the landscape, and when I use my wide-angle, the moon shows up as a dot. What the hell are people doing to get the moon so big?  Beginner skill level: I know this is the biggest culprit, but being the beginner, I'm unable to know where to adjust.