Metric induced by a surjective mapping Let $f$ be a continuous mapping of a compact metric space $(X, d)$ onto a Hausdorff space $(Y, \tau_1)$.  If $d$ is a metric on $X$, how to show that  $$d_1(y_1, y_2) := \inf\{d(a, b) : a \in f^{-1}(y_1)\text{ and }b \in f^{−1}(y_2)\}$$ is a metric on Y?  I can see that it is never $0$ when $y_1$ is not equal to $y_2$. But how should I go about showing that it satisfies the triangle inequality?  I am actually reading the following page from a book.
Metrizability of Hausdorff continuous image of compact metric space Let $f$ be a continuous mapping of a compact metric space $(X, d)$ onto a Hausdorff space $(Y, \tau_1)$. Then $(Y, \tau_1)$ is compact and metrizable.   In one proof the following metric is constructed:  $d_1(y_1, y_2) = inf\{d(a, b) : a \in f^{-1}\{y_1\},\ b \in f^{-1}\{y_2\}\}$, for all $y_1$ and $y_2$ in $Y$.  I'm thinking about how to prove that triangle inequality holds for $d_1$, s.t. $d_1(x, z) \leq d_1(x, y) + d_1(y, z)$  And how about the case when triangle inequality does not hold?    Update: @daniel-schepler suggested to use metric $d_1(x, y) = inf\{d(x^*, x_1) + d(y_1, x_2) + \cdots + d(y_{n-1}, x_n) + d(y_n, y^*)\}$ where $f(x^*) = x$, $f(y^*) = y$, and $f(x_i) = f(y_i)$ for each $i$.  Now we need to prove 3 property of metric: positivity, symmetry and triangle inequality. This is my sketch:   Positivity. Chose two points $x,y \in Y$ s.t. $x \neq y$. Singleton sets $\{x\}$ and $\{y\}$ are disjoint closed sets by Hausdorfness of $Y$ and their preimages are also disjoint closed in $X$ by continuity of $f$. $X$ is compact metric space by hypothesis and therefore a normal space. In normal space every two disjoint closed sets have disjoint open neighborhoods. Let $f^{-1}(x) \subseteq U$ and $f^{-1}(y) \subseteq V$ s.t. $U,V$ are open and $U \cap V = \varnothing$. Therefore there exists an $\varepsilon &gt; 0$, s.t. $x^* \in B_\varepsilon(x^*) \subseteq U$ and $y^* \in B_\varepsilon(y^*) \subseteq U$ and distance between any two closed sets is strictly positive and $d_1$ is positive and zero iff $x = y$. Symmetry. Clearly $d_1$ is symmetric. Triangle inequality. For triangle inequality we can show that in the "worst" case $d_1(x, z) = d_1(x,y) + d_1(y, z)$.
Can I change directory to a Windows drive in Ubuntu Bash on WSL? I am a super newbie for Ubuntu and basically need Ubuntu for a Python package.  I would like to run a Python file in say e:/username/folder1/folder2/python.py. I tried every method online and the only reply is bash: cd./e: No such file or directory. I am not sure whether it is because of the bash windows or virtualenv.  When I use bash on Ubuntu on Windows, ls, it says VIRTUALENV_DIR. cd, it says /home/username. The cd .. etc changes the directory a little, but not another drive, nor the folder I would like it to find. This integral is very closely related to the sophmores dream that states  $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$  For example here   Now I want to bound the integral, and showing that is less that 2.  For the interval $[0,1]$ a good bound is rewriting it to $\exp(x\log x)$ and using the expansion   $$ 1 - x \log(x) + \frac12 (-x \log(x))^2$$  but how does one handle $[1,\infty)$ ? In this answer here     gives bounds to the integral, but they are not tight enough.. So to taste my question again, how does one prove that   $$ \int_0^\infty \frac{\mathrm{d}x}{x^x} \leq 2 $$ I have noticed that the confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:    I initially thought this was because most values of the predictors were concentrated around the mean of the predictor. However, I then noticed that the narrow middle of the confidence interval would occur even if many values of were concentrated around the extremes of the predictor, as in the bottom left linear regression, which lots of values of the predictor are concentrated around the minimum of the predictor.  is anyone able to explain why confidence intervals for the predicted values in an linear regression tend to be narrow in the middle and fat at the extremes? I have a dual boot of Windows 10 and Ubuntu 16.04. Recently I upgraded it to 18.04. As soon as I did it, it started showing DISK IS LIKELY TO FAIL SOON (37° C / 99° F).    I checked the status of my disk via Windows and it shows it’s perfectly healthy. I assumed the error occurred because my partition size of Ubuntu was 64GB (which I now increased to 86GB). Please help me out with this.    Edit: The results of smartctl - attributes why $1+\frac{1}{1+\frac{1}{\frac{1}{1+\dots}}}$ cannot be equal to $\frac{1-\sqrt{5}}{2}$  I know that some of you will answer because it is positive.I know it but I don't know how to prove it.Maybe the second part is positive how do you know that?Thanks in advance. IOS 5.1.1 Safemode problem I jailbroke my old iPod Touch 3rd Generation, with IOS 5.1.1, using the latest version of absynthe. It worked for several months, but then it randomly stopped. It went into safe mode, but the touch screen wouldn't work. It showed the lockscreen, but nothing worked at all, except for the physical buttons. The only way to reboot is to follow the same steps as DFU mode (Hold power button 3 sec, then home button AND power for 10, then just home for 8)... However, it always boots right back into Safe mode.  I had to restore, and once rejailbroken, it worked fine, until I installed SBsettings. Then it did the EXACT same thing, and I had to restore and re-jailbreak AGAIN! The other day, I finished installing ALL of my extensive list of tweaks, and got it to about where I wanted it, when lo and behold, it booted to safe mode again. I have spent so much time on this, and I do not want to do it again. Does anybody here have any knowledge on the topic? Is there a program I can run from my computer to manually control the device or kick it out of safe mode? Is there some other method of doing this? I am looking for any solution OTHER than restoring. I have done that several times.  Also, I know the problem isn't with my jailbreak. It worked for several months before, then one day stopped. Now, it has the problem within a day of rejailbreaking, OR instantly if I install SBsettings. I re-installed the jailbreak, and applied the new one as well, and that didn't work. I have given up, and don't want to go through the painful process of restoring, rejailbreaking, and re-installing all of those tweaks again.  Also, because I can't get into my iPod, I can't provide a list of tweaks. I would if I could, but it just can't happen unless I can get in there. What are the rules of engagement with respect to SEO in HTML 5?  Examples of things we have been mulling over with no conclusion:   Are you penalised for not using &lt;head&gt; and &lt;body&gt; tags? Do links in &lt;nav&gt; blocks get prioritised for follows? With &lt;aside&gt; (no bside though), &lt;section&gt;, &lt;summary&gt; and &lt;details&gt; which ones carry the most weight for content, one-liner descriptions and keyword tags are there search engine implications? If you use CSS to put a &lt;footer&gt; element at the top and a &lt;header&gt; element at the bottom (not that you would but it may be construed as such by an incomplete spider implementation)? Does the use of placeholder attributes on &lt;input&gt; show you are more user focused?   At the minute we are haphazardly pinning the content on the donkey with regard to tag usage but would be interesting to know if anyone has any insight into the "correct" tags to use with regards to SEO. Prevent Photos app on OSX from launching when plugging in device The Photos app opens whenever I plug in my phone.    How can this be prevented?  I have seen many solutions for the old iPhoto app that involve changing the "Connecting camera opens" option, but the new app has no such option. 3d transforms on SVG element Use grep --exclude/--include syntax to not grep through certain files I'm looking for the string foo= in text files in a directory tree. It's on a common Linux machine, I have bash shell: grep -ircl &quot;foo=&quot; *  In the directories are also many binary files which match &quot;foo=&quot;. As these results are not relevant and slow down the search, I want grep to skip searching these files (mostly JPEG and PNG images). How would I do that? I know there are the --exclude=PATTERN and --include=PATTERN options, but what is the pattern format? The man page of grep says: --include=PATTERN     Recurse in directories only searching file matching PATTERN. --exclude=PATTERN     Recurse in directories skip file matching PATTERN.  Searching on grep include, grep include exclude, grep exclude and variants did not find anything relevant If there's a better way of grepping only in certain files, I'm all for it; moving the offending files is not an option. I can't search only certain directories (the directory structure is a big mess, with everything everywhere). Also, I can't install anything, so I have to do with common tools (like grep or the suggested find). Footnotes in tables? I would like to have footnotes in tables, but the \footnote command does not work inside a table environment.  Googling one finds that this is , at least in the UK.  That link contains a list of solutions to the problem, but since the author of the FAQ does not believe that this is good style, no comparison is made between the different methods.  I am sure that there are good typographical reasons why in most situations it is undesirable to have footnotes in tables (although I have never come across them), however I have recently felt the need to do just this in a paper I am currently writing.  Therefore I'd like to ask two questions:   What are the arguments against footnotes in tables? If one really must go against the advice in 1., which method (whether considered in the  or not) would you recommend?   Thanks in advance.  Added  In response to ShreevatsaR's comment: I do not much care for where the text of the footnote appears.  I do however insist on the footnote marks being attached to the relevant entries in the table, so that I can avoid having to reference "the second entry in the eighth line of the table", etc... How to insert strings containing slashes with sed? Orthogonal Projection of $ z $ onto the Affine set $ \left\{ x \mid A x = b \right\} $ How do I report a bug related to Stack Exchange? Does Stack Exchange contain any edit suggestion or bug submission system, or such type of discussion forum to inform SO or its administrators about suggestions and bug reports that relate to SO itself? How can I stop Lollipop ignoring my wifi network? Android 5.0 has a new "feature": If a wifi network has no connection to the internet, it will remain connected to it but route traffic over mobile data instead. A small exclamation mark shows in the status bar over the wifi symbol.  Unfortunately, for some reason Android has decided that my home wifi does not have a good internet connection - which is not true, as I'm using it to access Stackexchange at the moment - and is, instead, trying to send everything over GPRS.  With mobile data disabled, things work over wifi just fine, despite the exclamation mark - so it isn't a problem with the wifi not working, per se.  Is there any way to override this "smart" feature? The only way I have found to make things work at the moment is to disable mobile data while I am in my house - which is not a practical long-term solution.   EDITING to add that I have now solved this problem by buying a new router! I was using something very old... But, I'll leave the question here for the benefit of anybody else with the same difficulty. Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$ All the units are satisfied Pell's equation $a^2-2b^2=\pm1$  for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$.  Here is my proof:  Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies $$(a+b\sqrt{2})(c+d\sqrt{2})=1, c+d\sqrt{2}\in\mathbf{Z}\sqrt{2}$$ $$\implies \mathrm{norm}((a+b\sqrt{2})(c+d\sqrt{2}))=1$$ $$\implies \mathrm{norm}(a+b\sqrt{2})\cdot \mathrm{norm}(c+d\sqrt{2})=1$$ $$\implies (a+b\sqrt{2})(a-b\sqrt{2})(c+d\sqrt{2})(c-d\sqrt{2})=1$$ $$\implies (a^2-2b^2)(c^2-2d^2)=1$$ $$\implies \left[ a^2-2b^2=1\text{ and } {c}^2-2{d}^2=1\right]\text{ or } \left[ a^2-2{b}^2=-1\text{ and }{c}^2-2{d}^2=-1\right]$$ $$\implies {a}^2-2{b}^2=\pm1$$ How do I install Ubuntu from an USB drive / create a live USB-stick in Windows? I tried to use the usb-creator.exe from Ubuntu 10.10 desktop i386 ISO image and it does create an USB stick that doesn't boot. When I try to boot from it I get:     Operating System not found.   I try to do this from Windows 7 x64. I'm not looking to create a live Ubuntu media, only an installation USB disk.  I repeated the process several times and I even tried to reformat the USB drive (FAT32), no success yet.  The USB stick is a 8GB one. How to make clickable links and cross-references in pdfs produced by LaTeX? I wrote a Table Valued Function in Microsoft SQL Server 2008 to take a comma delimited column in a database to spit out separate rows for each value.  Ex: "one,two,three,four" would return a new table with only one column containing the following values:  one two three four   Does this code look error prone to you guys? When I test it with   SELECT * FROM utvf_Split('one,two,three,four',',')    it just runs forever and never returns anything. This is getting really disheartening especially since there are no built in split functions on MSSQL server (WHY WHY WHY?!) and all the similar functions I've found on the web are absolute trash or simply irrelevant to what I'm trying to do.  Here is the function:  USE *myDBname* GO SET ANSI_NULLS ON GO SET QUOTED_IDENTIFIER ON GO ALTER FUNCTION [dbo].[utvf_SPlit] (@String VARCHAR(MAX), @delimiter CHAR)  RETURNS @SplitValues TABLE (     Asset_ID VARCHAR(MAX) NOT NULL )  AS BEGIN             DECLARE @FoundIndex INT             DECLARE @ReturnValue VARCHAR(MAX)              SET @FoundIndex = CHARINDEX(@delimiter, @String)              WHILE (@FoundIndex &lt;&gt; 0)             BEGIN                   DECLARE @NextFoundIndex INT                   SET @NextFoundIndex = CHARINDEX(@delimiter, @String, @FoundIndex+1)                   SET @ReturnValue = SUBSTRING(@String, @FoundIndex,@NextFoundIndex-@FoundIndex)                   SET @FoundIndex = CHARINDEX(@delimiter, @String)                   INSERT @SplitValues (Asset_ID) VALUES (@ReturnValue)             END              RETURN END How to draw commutative diagrams? Can anyone help me in giving commands for commutative diagrams in LaTeX ? It doesn't seem to appear properly in the LaTeX guides that I have. A command for a simple triangular diagram with arrows associated to maps would be good enough, or maybe some appropriate reference.