Show that the plane with countably many points removed is path-connected under the Euclidean topology. This is quite an interesting issue, hopefully everyone will help you
Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected?  This is not homework but is in response to working several problems where countable or uncountable many points are removed and the pattern always holds.  Thank you.
There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. What is the difference between "speaking" and "talking"? It seems more politically correct to say that someone is speaking rather than talking.  What is the definitive difference between these terms? Does the VBA "And" operator evaluate the second argument when the first is false? This is on Stack Overflow. On Meta I'm able to see what the tags are on the right side of the screen. In fact, I think yesterday or some point earlier today I saw tags on the right side of the screen (I can't screenshot or remember correctly).  It's right above the asked and active heading.  I know this is minor but what happened to it? It might be useful for browsing instead of having to look at the bottom of the question post.    Found an image on Google and it shows the tags on the right side of the screen: MLE for Uniform $(0,\theta)$ I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.  I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood function.  What is confusing me is that if a function is decreasing, then wouldn't the function be maximized at the smallest input rather than the largest?  Thank you in advance for your help. Object of class mysqli_result could not be converted to string Is there a way that I can configure Visual Studio 2008 to understand CamelCase?  Specifically, I'd like to be able to get Ctrl + right or left cursor to take me to a subsection of a variable or type name. i.e., if my cursor was at the start of this line: LongNamedExampleClass longNamed = new LongNamedExampleClass();  and I hit Ctrl + right, I'd like to jump to &quot;Named&quot; in the class name, rather than the start of the variable name.  If I hit it again, I'd like to go to &quot;Example&quot;.  I hope that makes sense. I used this navigational feature in Eclipse quite a lot, and find it difficult to unlearn something so ingrained. Need help with proof for Dedekind cuts on $\mathbb{Q}^+$ I am working on a proof about Dedekind cuts on the positive rational numbers. I have been stuck for a while on the following point and would appreciate any help.  Given $x\in \mathbb{Q}^+$ such that $x^2&lt;2$, how to find $y\in \mathbb{Q}^+$ such $y&gt;x$ and $y^2&lt;2$? Explaining to an 8-year-old why nothing can travel faster than light How to remove top and bottom whitespace of longtable? I am using longtable instead of tabular environment and noticed that it puts some additional whitespace at the top and at the bottom of the table. Is there some way to remove it (something less hacky than \vskip -\baselineskip before the table)?  Small piece of code for illustration:  \documentclass[12pt]{book} \usepackage{longtable}  \begin{document}  Linelineline  \begin{longtable}{l l} cell1 &amp; cell2 \end{longtable}  Linelineline  \begin{tabular}{l l} cell1 &amp; cell2 \end{tabular}  Linelineline  \end{document}   And that's what I am getting:    Common tabular environment does not have any additional margins, but longtable does. The difference between two periodic functions converges to zero, is this two functions identical? If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in \Bbb R$. Now that $\displaystyle\lim_{x\to\infty}(f(x)-g(x))=0$.  Conjecture: $f(x) \equiv g(x)$. Is it possible with GEE to export all the images in an image collection? Audio is working, but the speaker test doesn't work I've installed Ubuntu 12.04 LTS from a minimal cd on a netbook (Asus 1001 PXD). I've installed the ubuntu-desktop package using the --no-install-recommends option.  Everything works fine, except the "sound test" for headphones or analog speakers.  Clicking on the "test" buttons (front left and front right) I don't hear any sound. Despite this, the audio is working properly.  I've checked the audio levels with alsamixer; I have also checked that the test sounds actually exist in /usr/share/sounds/alsa.  I tried an installation of Ubuntu 12.04 LTS made with the desktop-cd, and in this case the speaker test works properly.  I suppose, therefore, that the problem could depend on the lack of a package, but I have not identified which one. Is it possible to paint only inside a stroke on illustrator? what i actually want to do is paint the same stroke with different colors in diferent segments of it but i can't do it...   Also... I've seen some tutorials that include editing a gradient but that would be very complicated and so would be to make a new stroke over this one, I just hope theres an easyer way XD  Edit per comment:  thats cool but not really what im looking for, what i want is lets say my stroke has the shape of a burger i want part of i to be bread color and part of it to be meat color... not one stroke over the other... i want something like this, imagine the jelly worm is my stroke: How to disk copy with unreadable sectors? I want to clone a whole disk bytewise, something like  dd if=/dev/$SRC of=/dev/$DST bs=65536 count=$count   There are better tools for the job, which understand the file system (Windows) and work faster (I don't care), but they fail because of unreadable sectors on the old source disk.  There's actually a single invalid block I have to skip. So I thought about using dd multiple times like  dd if=/dev/$SRC of=/dev/$DST bs=65536 count=... dd if=/dev/zero of=/dev/$DST bs=65536 count=... skip=... seek=... dd if=/dev/$SRC of=/dev/$DST bs=65536 count=... skip=... seek=...   with replacing the broken sector data by zeros. Is this a valid approach? I found this  which confuses me. Prove $\sup(f+g) \le \sup f + \sup g$ Suppose $D$ is a nonempty bounded subset of reals. Let $f:D \to \mathbb R$ and $g:D \to \mathbb R$. Define $(f+g)(x)=f(x)+g(x)$. Prove $\sup(f+g)(D) \le \sup f(D) + \sup g(D)$ (also prove that $\sup (f+g)$ exists).   I understand why this is the case, just not how to prove it. Left side is pretty much $\sup  (f(x)+g(x))$ and right side is $\sup (f(x) + g(y))$ for $x,\,y \in D$. Basically $f+g$ has to use the same variable and $f(D)+g(D)$ use different ones. But I don't know how to go about proving this.     The second part of the question is to find a specific example where strict inequality holds.    Let $D=\{a,b\}$ and $f: a \to 1,\, b\to 0, \,g: a \to 0,\, b\to 1$. $\sup f(D) = 1,\, \sup g(D) = 1,\, \sup f(D) + \sup g(D) = 2.$ $\sup (f+g)(D) = 1$ (if we choose a, $f+g = 1+0,\, b,\, f+g=0+1$). What is the difference between "I have a doubt with ...." & "I suspect there is a problem"? I used the following code to list my java code, but the it looks like there's too much space between words. What can I do?  % source code listing \usepackage{listings} \usepackage{color} \usepackage{xcolor}  \usepackage{caption}  % citation style \usepackage{apacite} \DeclareCaptionFont{white}{\color{white}} \DeclareCaptionFormat{listing}{\colorbox{gray}{\parbox{\textwidth}{#1#2#3}}}  \begin{document} \begin{lstlisting}[label=some-code,caption=bind to service call] bindService(new Intent(this, Recognizer.class), recConnection, Context.BIND_AUTO_CREATE); \end{lstlisting} \end{document}   The snapshot of what I got. The code not completely shown, and space too wide, in function bindService. Distribution of range of Uniform $(0,1)$ distribution Why is "Nature" usually spelt with an initial capital letter in scientific journals? I am mainly referring to life science here, in case this matters.  I am not talking about the obvious cases, like capitalisation in titles, at the beginning of a sentence, etc., but when it appears mid-sentence, where one would normally expect it to be lowercase.  I've heard some explanation that it gives the impression of Nature as a goddess (see comments on this post ) - but this is probably the last thing I would have expected in a scientific article...