Show that a topological space $X$ is separated (Hausdorff) if and only if for all compact subspaces $K_1$ and $K_2$ of $X$, there are disjoint sets $U$ and $V$ of $X$ such that $K_1 \subseteq U$ and $K_2 \subseteq V$.  I have one way ($\Leftarrow$), but I'm having some problems finding a neat proof from left to right. I think I've found a solution, but it's quite tedious. Here it is:  We assume $X$ is separated and we take two disjoint compact subspaces $K_1$ and $K_2$ of $X$.  Let's fix $x \in K_1$. Then $\forall y \in K_2$, there are two disjoint open sets $V_{x,y}$ and $U_{x,y}$ including $x$ and $y$ respectively. All these $U_{x,y}$ cover $K_2$ and therefore we can extract a finite number of them that cover it too $\{U_{x,y_i} : i = 1,...,n\}$ ($n \in \mathbb{N}^{\ast}$), and we take the correspondent $\{V_{x,y_i} : i = 1,...,n\}$. Then the sets $V_x:= \bigcap_{i = 1}^n V_{x,y_i}$ and $U_x := \bigcup_{i = 1}^n U_{x,y_i}$ are open disjoint sets.  Now I do this for all $x \in K_1$ and get a set $\{V_x : x \in K_1\}$ of open sets covering $K_1$, and corresponding $\{U_x : x \in K_1\}$ such that $\forall x \in K_1, V_x \cap U_x = \emptyset$. Since $K_1$ is compact, I now extract a finite number of those open sets $\{V_{x_i}: i=1,...,m\}$ ($m \in \mathbb{N}$), which still cover $K_1$ and the corresponding $\{U_{x_i}: i=1,...,m\}$.  Now $V :=\bigcup_{i = 1}^m V_{x_i}$ is open and covers $K_1$. While $U :=\bigcap_{i = 1}^m U_{x_i}$ is open too and covers $K_2$. Moreover $U \cap V = \emptyset$. That finishes the proof.  Is this approach correct? But more importantly, isn't there a less tedious way?
I want to show that any two disjoint compact sets in a Hausdorff space $X$ can be separated by disjoint open sets. Can you please let me know if the following is correct? Not for homework, just studying for a midterm. I'm trying to improve my writing too.  My work:  Let $C$,$D$ be disjoint compact sets in a Hausdorff space $X$. Now fix $y \in D$ and for each $x \in C$ we can find (using Hausdorffness) disjoint open sets $U_{x}(y)$ and $V_{x}(y)$ such that $x \in U_{x}(y)$ and $y \in V_{x}(y)$. Now the collection $\{U_{x}: x \in C\}$ covers $C$ so by compactness we can find some natural k such that  $C \subseteq \bigcup_{i=1}^{k} U_{x_{i}}(y)$   Now for simplicity let $U = \bigcup_{i=1}^{k} U_{x_{i}}(y)$, then $C \subseteq U$ and let $W(y) = \bigcap_{i=1}^{k} V_{x_{i}}(y)$. Then $W(y)$ is a neighborhood of $y$ and disjoint from $U$.  Now consider the collection $\{W(y): y \in D\}$, this covers D so by compactness we can find some natural q such that $D \subseteq \bigcup_{j=1}^{q} W_{y_{j}}$.  Finally set $V = \bigcup_{j=1}^{q} W_{y_{j}}$, then $U$ and $V$ are disjoint open sets containing $C$ and $D$ respectively.   What do you think?
Is there any freely available pointcloud data?   I'm especially interested in dense clouds of small areas(5m to 100m radius) but anything else would be fine too.   I've found that these pages have very nice and easy to access point cloud data. What are the criteria used to represent Stack Exchange sites on the sites page? (Grid view) I noticed that  has been changed, and by default sites are shown by balloons.  What is not clear to me is why some sites are shown with a bigger balloon than other sites. For example, WordPress Answers has less questions, answers and users than Programmers, but its balloon is bigger. What are the criteria used to show the sites on that page? Why does python 2.5.2 have the following behavior  &gt;&gt;&gt;[2].extend([]) == [2] False  &gt;&gt;&gt; [2].extend([]) == None True  $ python --version Python 2.5.2   I assume I'm not understanding something here, but intuitively I'd think that [2].extend([]) should yield [2] I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation  $p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$  I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).   Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood? Einstein-Hilbert action as an effective field theory I have read the following statement:      "The Einstein-Hilbert action can be seen as an effective field theory: setting $g_{\mu\nu}= \eta_{\mu\nu}+h_{\mu\nu}$, one gets a free Lagrangian for the field $h_{\mu\nu}$ plus interactions."   Now, how is it possible to define all the interaction terms?  EDIT: I ask apologize if my statement was appeared unclear: I have created a  which is more precise; please don't continue to consider seriously this one. Adding coordinate system to layout in QGIS composer? How to use WiFi and Ethernet at the same time Brief Description:  I need my Ethernet card (en0) to see all traffic under 192.168.2.xxx.  However I also need to use WiFi card (en1) for all other traffic.  OS: OS X Lion    I am using a separate network location to deal with this specific use (since this is a for a robotics project that communicates via Ethernet). Ethernet (en0) had to be set up as a static IP (192.168.2.10) per the requirements for the device I am connecting to. All connections on the Ethernet card (en0) should be 192.168.2.xxx as specified above. WiFi (en1) is set up as DHCP to a router since their is no point in assigning it as a static for general purpose traffic. Ethernet (en0) is set up as priority over WiFi (en1).  I would also like to keep this under the network location preferences if at all possible since I do not require this feature to be on all the time.  I would really appreciate your help on this. No one I have talked to knows how to solve this. delay JQuery effects I want to fade out an element and all its child elements after a delay of a few seconds. but I haven't found a way to specify that an effect should start after a specified time delay. Prove the existence of an analytic function Let $\Delta=\{z\in \mathbb C: |z| &lt; 1\}$. Let $f: \Delta \rightarrow \mathbb C$ be a one-to-one analytic function fixing the origin. Prove that there is a one-to-one analytic function $g: \Delta \rightarrow \mathbb C$ such that $[g(z)]^2=f(z^2)$. Further, show that such a function is odd.  The only idea which comes to my mind is the following (not sure whether it is the right direction though). Let $h: \Delta \rightarrow \Delta \rightarrow \mathbb C,\ z\mapsto z^2\mapsto f(z^2)$. Note that $h'(0)=0,\ h''(z)=2f'(z^2)+4z^3f''(z^2)$; since $f$ is one-to-one and analytic, its derivative never vanishes, so $h''(0)\ne 0$. So $h$ is an analytic function of $\Delta$ with a zero of order two at the origin. Then there is an analytic function $g$ defined in an open disc about the origin with a simple zero at the origin such that $[g(z)]^2=f(z^2)$.   This is the only thing I came up with, but it seems to be a little bit irrelevant since I don't see any way of extending this $g$ to the whole $\Delta$, and also the injectivity and oddness of such a $g$ are unclear. I have a really strange situation here. My PC works fine, at least in most cases, but there's one thing that I can't deal with. When I try to copy a file from my pendrive, everything is ok -- I got 16-19M/s , it works pretty well. But when I try to copy something to the same pendrive, my PC freezes. The mouse pointer stops moving for a sec or two, then it moves a little bit and it stops again. When something is playing, for example, in Amarok, the sound acts like a machine gun. The speed jumps from 500K/s to 15M/s, average 8M/s. This occurs only when I'm copying something to a pendrive. When the process of copying is done, everything backs to normal.   I tried everything -- other pendrive, a different USB port on front panel or those ports from back, I even changed the USB pins on motherboard (front panel), but no matter where I put my USB stick, it's always the same. I tried different filesystem -- fat32, ext4. I have no problem with the device on Windows, on my laptop. It has to be my PC or something in my system. I have no idea what to look for. I'm using Debian testing with standalone Openbox. My PC is kind of old -- Pentium D 3GHz, 1GiB of RAM, 1,5TB WD Green disk. If you have something that would help me to solve this issue, I'd be glad to hear that.  I don't know what else info I should provide, but if you need something, just ask, I'll update this post as soon as possible.  I tried to reproduce this problem on ubuntu 13.04 live cd. I mounted my encrypted partition + encrypted swap and connected my pendrive to a usb port. Next I tried to start some apps, and now I have ~820MiB in RAM and about 400MiB in SWAP. There's no problem with copying, no freezing at all, everything is as it should be. So, it looks like it's a fault of the system, but where exactly? What would cause such a weird behavior? Converting a list of tuples into a dict I have a list of tuples like this:  [ ('a', 1), ('a', 2), ('a', 3), ('b', 1), ('b', 2), ('c', 1), ]   I want to iterate through this keying by the first item, so, for example, I could print something like this:  a 1 2 3 b 1 2 c 1   How would I go about doing this without keeping an item to track whether the first item is the same as I loop around the tuples? 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Problem:   Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product   My attempt:  This problem would be solved if it could be proven that 9 consecutive numbers cannot exist such that all the numbers are of the form $2^a  3^b  5^c  7^d$ where $a,b,c,d$ are integers.  A suitable hint would be appreciated, I don not want a complete solution.  Edit:  If there existed a prime factor greater than $7$ of any number present in the set, then its multiple cannot exist b/c $22-11&gt;9$  Edit 2: As mentioned by Erick, my reasoning is not valid for numbers $1-9$ and $2-9$, let this be an exception. How to get a DOM Element from a jQuery selector? Dissolve (merge) polygons QGIS based on attribute values only when are adjacent Get the name of an object's type