Set S=Q$\cap$(0,1) I wanted to show that this can not be expressed as intersection of countable collection of open sets. I know Any closed set can be shown as intersection of countable collection of open set .I had hint show this by Cantor intersection property which says that infinite intersection of  nested sequence of nonempty compact set is nonempty . I could not able to link this hint with problem till now .Any help will be appreciated    Without using Baire Category Theorem Is it possible to solve this problem? As this problem occur in exercise where there in no mention of that theorem in text.
I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks!
Is there any hint of of the first Wizarding War being an international event? Does each season of Game of Thrones roughly equal a book? Does each current Game of Thrones season roughly equal one A Song of Ice and Fire book? And do we know which future seasons will cover which books?  I am planning to read the first book after the TV show has finished depicting the events in the first book. I would like to know if after each TV season ends I can expect to be able to read a book covering the events depicted on screen? What is the Physical Meaning of Commutation of Two Operators? I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (self-adjoint operator) in quantum mechanics?   E.g. an operator $A$ with the Hamiltonian $H$? I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? Bridged adapter in Virtual Box - cannot obtain IP I have virtual box with RHEL installed. My virtual box is running on windows 7 64 -bit. I configured bridged connection, but when I am restarting network interfaces, I receive following error:  Determining IP information for eth1... failed   On Windows XP it works without problems. I disabled Windows Firewall and installed latest Virtual Box, still without success.   NAT is working without any problems. What did I missed?  UPDATE  Ouput of ifconfig -a is:  eth0      Link encap:Ethernet  HWaddr 08:00:27:B6:FC:69           inet addr:10.0.2.15  Bcast:10.0.2.255  Mask:255.255.255.0           inet6 addr: fe80::a00:27ff:feb6:fc69/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:430 errors:0 dropped:0 overruns:0 frame:0           TX packets:346 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:36520 (35.6 KiB)  TX bytes:31455 (30.7 KiB)  eth1      Link encap:Ethernet  HWaddr 08:00:27:C3:84:4A           inet6 addr: fe80::a00:27ff:fec3:844a/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:1464 errors:0 dropped:0 overruns:0 frame:0           TX packets:17 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:135419 (132.2 KiB)  TX bytes:4230 (4.1 KiB)  lo        Link encap:Local Loopback           inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:16436  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)   Output of route -n  Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 10.0.2.0        0.0.0.0         255.255.255.0   U     0      0        0 eth0 169.254.0.0     0.0.0.0         255.255.0.0     U     1002   0        0 eth0 0.0.0.0         10.0.2.2        0.0.0.0         UG    0      0        0 eth0   Output of dhclient -v eth1 is:  Listening on LPF/eth1/08:00:27:c3:84:4a Sending on   LPF/eth1/08:00:27:c3:84:4a Sending on   Socket/fallback DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 6 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 9 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 15 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 18 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 13 (xid=0x6afb6c4c) No DHCPOFFERS received. No working leases in persistent database - sleeping.   Moreover the same machine with the same configuration works well on Windows XP. The problem is only on Windows 7. Generate the smallest $\sigma$-algebra containing a given family of sets My teacher gave me an example of performing the subject:     Example      Let $\Omega = \Bbb R$ and $\mathcal R = \{(-\infty,-1),(1,+\infty)\}$.   Then $\sigma(\mathcal R) = \{\emptyset, \Bbb R, (-\infty,-1),  (1,+\infty), [-1, \infty), (-\infty,1],  (-\infty,-1)\cup(1,+\infty),[-1,1]\}$.   There was a different example, where she also generated the smallest  $\sigma$-algebra for the family of sets $\mathcal A = \{A,B\} \subset 2^\Omega$ in the same way: $\sigma(\mathcal R) =  \{\emptyset, \Omega, A, B, A^C,B^C,A\cup B, (A\cup B)^C\}$.   I certainly understand why $\emptyset$, $\Omega $ and a family of sets itself are there, and I certainly know that $\sigma$-algebra is closed under the operations of complement and union. What I don't understand about the other elements of generated $\sigma$-algebras: why we are taking exactly them? Does this work in a general case? Copy skype chat history to text files? I'd like to convert my Skype chat history to a collection of text files for archival purposes.  Ideally, I'd like to have a command (e.g. script or app) so that I could do this on a regular basis.  Any one have any idea on how to go about doing this? Adding Layer Files (.lyr, .lyrx) to QGIS At college we use ArcGIS Desktop 10, but that does not function on my Mac so I have QGIS there instead.  I want to do my exercises with QGIS, but in ArcGIS Desktop we use .lyr files with ArcMap and .lyrx files with ArcGIS Pro, and they are not supported by QGIS.  Does anyone know, how I can connect the *.lyr files to my *.shp files to use them on my Mac with QGIS? How do I convince gdal2tiles to output pngs with transparency? Tikz won't plot the function properly I tried to plot the following function but it didn't work:  y = 300 + 1000/x + 2.5x.  This is the document, but instead of the plot it only shows the axis.  \documentclass{article}  \usepackage{pgfplots}% This uses tikz \pgfplotsset{compat=newest}% use newest version  \pgfmathdeclarefunction{Function}{1}{%   \pgfmathparse{300 + 1000/x + 2.5}% }  \tikzset{My Line Style/.style={smooth, thick, samples=400}}  \begin{document} \begin{tikzpicture} \begin{axis}[     xmin=0.0,     xmax=250,     ymin=0,     ymax=50,     axis x line=bottom,     axis y line=left,      xlabel= $y$,     ylabel=Cost,      xtick={250},     xticklabels={250},         restrict x to domain=0:250,     ]        \addplot+[My Line Style, color=black, samples at={1,2,3,4,...,249,250}] (\x,{Function(\x)});   \end{axis}  \end{tikzpicture} \end{document} \documentclass{article} Why is ++i considered an l-value, but i++ is not? Why is ++i is l-value and i++ not? Is it appropriate to buy a "thank you" gift for a PhD supervisor? I am nearing the end of my PhD and I would quite like to buy a "thank you" gift for my supervisor (although, she's the socially awkward type who I'm sure will be greatly embarrassed to receive one).  I'm just wondering whether a gift is the done thing? And if so, what sort of gift seems appropriate? I suppose a bottle of wine would be a safe choice? How to create & execute a script file Debug a java application without starting the JVM with debug arguments Normally to attach a debuger to a running jvm you would need start the jvm with arguments such as the following:  &gt; java -Xdebug -Xrunjdwp:transport=dt_socket,address=1000,server=y,suspend=n   Now if I want to debug a process that wasn't started in debug mode, what can I do?  This situation arrises when a production system (i.e. started without debug args) exhibits a 'random' (I use the term loosely) bug. So I can't restart the jvm with the appropriate arguments, because nobody knows how to reproduce the bug again. Is it impossible to attach to the JVM in this situation?  Just to clarify it is not possible to use tools like jdb to attach to already running JVMs unless they were started in debug mode  from the JVM man page     Another way to use jdb is by attaching it to a Java VM that is   already running. A VM that is to be   debugged with jdb must be started with   the following options: If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? Any help would be appreciated.  If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? Two git repositories in one directory? Is it possible to have 2 git repositories in one directory?  I'd think not, but thought I'd ask.  Basically, I'd like to check in my home directory config files (e.g. .emacs) which should be common across all of the machines I work on, but have a second repository for local files (e.g. .emacs.local), which contains machine-specific configurations.  The only way I can think of to do that is to have the local config in a subdirectory and ignore that subdirectory from the main git repository.  Any other ideas? Every time I render an image and try to save it, it just disappears. How do you exactly save a render? I am using a mac. Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) \, dx $ Assume we have:  $$ \int (\cos{x} + \sin{x}\cos{x}) \, dx$$  Two ways to do it:  Use $$\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $$ Then $$ \int \left(\cos{x} + \frac{\sin{2x}}{2} \right) \, dx = \sin{x} - \frac{ \cos{2x} }{ 4 } + C. $$ The other way, just see that $ u = \sin(x), du = \cos(x)dx $, $$ \int ( \cos{x} + \sin{x}\cos{x}) \, dx = \sin{x} + \frac{\sin^2{x}}{2} + C. $$  Now the part I don't see fully is, why aren't these results completely equal?  Taking the 2nd result, \begin{align} \sin{x} + \frac{\sin^2{x}}{2} &amp;= \sin{x} + \frac{1}{2} \, \left( \frac{1 - \cos{2x}}{2} \right) \\ &amp;= \sin{x} + \frac{1}{4} - \frac{\cos{2x}}{4}. \end{align}  So you have to absorb $\frac{1}{4}$ into $C$ for them to be equal.  Shouldn't they be equal straight away? Finding the maximum likelihood estimators for this shifted exponential PDF? Fibonacci sequence and Euclidean algorithm's connection. Let $f_1, f_2,\dots,f_i,\dots$ be the Fibonacci sequence with $f_1=f_2=1$. For any $a,b\in\mathbb{N}$ with $a&lt;b$, suppose the Euclidean algorithm takes $n$ long divisions to compute $\gcd(a,b)$. If $a\le f_j$, then $n\le j-1$, for $j\ge2$.   I want to prove this problem. But I can't see any connection between Fibonacci and Euclidean algorithm. If anyone can give a hint about first step, it will be very thankful.  My guess is to use induction.