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Homeomorphism of the real line-Topology Normal + Connected -> Uncountable Here's a solution (to the exercise "prove that if $X$ is a space with more than one element, and is normal and connected then $X$ is uncountable"):   by Urysohn's lemma, given $A$ and $B$, closed and disjoint in $X$, there exists a continuous function from $X$ into $[0,1]$ such that $f(A)={0}, f(B)={1}$.   If $X$ were countable, so it would be $f(X)\subset [0,1]$; choose $r\in (0,1)\setminus f(X)$, then $X=f^{-1}([0,r))\cup f^{-1}((r,1])$, so $X$ is disconnected.  But the full power of the lemma wasn't used, all that was used is that there exists a continuous function $X\longrightarrow [0,1]$, so the condition of being normal seems too much.  So my question is, what are the weakest conditions on $X$ for the existence of such continuous functions into $[0,1]$ (or $\mathbb{R}$, for that matter)? Smeared creamy look effect i came across a photo where the gradient looks like it s smeared from cream. It is hard to explained by saying so here is the  Let say i create 2 similar shapes with the gradient pattern then adjust the opacity but the edges will still remain. But in the example the edges are appeared to be soft and hard randomly. Thanks in advance ! I find myself needing to draw lots of elegantly curved paths in TikZ. Ideally, I'd just specify a series of points, and TikZ would calculate the extra data itself to draw a nice series of curves passing smoothly through these points, perhaps with an optional "looseness" parameter that I could specify. But the only way I can find to draw nice curves is by explicitly giving control points, or by manually specifying in and out angles.  I can think up a simple algorithm to do this, which would certainly be within TikZ's power to perform: just choose the in and out angles in a simple fashion based on the relative angles between each adjacent pair of line segments.  Is something like this already built-in? Or can someone cook something up that does the job?  Edit: Jake has provided an answer using the plot [smooth] functionality. This is almost perfect! But it can't do what I need, because it doesn't let me specify tangent angles manually where needed, which is especially important at the beginning and end of the curve. I would have thought this would be a natural and straightforward addition to the existing plot [smooth] algorithm: for every coordinate, an optional angle should be able to be specified as an argument, which if supplied is treated as the tangent angle for the curve at that point. And while we're at it, it wouldn't hurt to also allow the tension to be modified along the path.  A minimal extension to the algorithm would just accept two optional parameters, for the curve tangent at the beginning and end. I'm new to Linux and Ubuntu. People keep using ~/filename in their answers to my questions. What does ~/ mean? I just installed Ubuntu 12.10 (amd64), and want to install a recent version of MySQL 5.6.  If possible, I would like to install (not upgrade) it the "Debian Way' (i.e., using apt-get or dpkg). Synthetic division of polynomials by factor of the form $(ax+b)$ How can I disable Alt-Scroll zoom in XFCE4 (Xubuntu) In XFCE4 desktop environments, alt+scroll wheel zooms my entire desktop. I'm finding this very annoying because sometimes I flick the scroll wheel to go to the top of a document then press the alt key for a some unrelated keyboard shortcut and find myself zoomed in so far I can't see anything at all. Is there any way to disable this feature? 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 $n\in \mathbb{N}$. Two icons on Unity Launcher on non-standard application launch I have an application which doesn't come from any Ubuntu repo (neither official, nor PPA) but available as tgz and supposed to be unpacked and ready to use. To be precise it is .   The problem is the  application when launched shows its own icon in Unity Launcher. After making a custom .desktop file and placing shortcut onto Unity Launcher (by dragging from the dash) and launching it I see two icons - one placed by me and another one with real application.  This is the .desktop contents:  [Desktop Entry] Version=1.0 Type=Application Terminal=false Exec=/opt/LightTable/LightTable Name=Light Table Icon=/opt/LightTable/core/img/lticon.png   One more thing. The /opt/LightTable/LightTable is bash script file and eventually /opt/LightTable/ltbin is being launched. I think this is the problem but have no idea how to fix the problem. How can I avoid asking duplicate questions? If the distribution of test statistic is bimodal, does p-value mean anything? Is it possible to be admitted to a masters in Germany with a 3-year bachelors from another country? Should we have a subjective version of every Stack Exchange site? Can a particle be physically observed inside a quantum barrier? 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? Below is experimental data for the ratio $$R=\frac{\sigma(e^+e^-\rightarrow hadrons)}{\sigma(e^+e^-\rightarrow\mu^+\mu^-)}$$ as a function of the centre of mass energy $\sqrt s$. I am interested in the peak at around $100GeV$ which corresponds to the resonance of the $Z^0$ boson. There are two ways of looking at this that I have in mind:  We can effectively ignore the electromagnetic process around this energy, and so each process should have it's own Breit-Wigner peak centred on the mass energy of the $Z^0$. The ratio of these is just a constant. Thinking in terms of Feynman rules and again ignoring the electromagnetic process around this region, each process has the same propagator and vertex factors (roughly - ignore quark mixing) and there are some extra factors due to different quarks being possible and colour degeneracy, but still the ratio should be a constant (one possible issue here is interference between Feynman diagrams which I have neglected?).  So my question is why does the peak exist in the data? The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ (NBHM_2006_PhD Screening Test_Analysis)     The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the   point on the curve farthest from the line $x = y$   how should i solve this problem Does Auto White-Balance Really Work? How? I just learned that using CBC encryption with an IV which is predictable is not secure.  From what I understand, using certain plain texts, and then guessing the IV that it uses, the attacker can verify if the IV he guessed was right. How does this qualify as an attack? I am putting together a GIS server using GeoServer and need some help with importing my layer styles from ArcGIS 10 to  format.    I have one layer in ArcMap properly symbolized and labeled, how can I get this layers symbolization and labeling into a SLD file?