There exists a well-ordered set $A$ having a largest element $\Omega$ such that the section $S_\Omega$ of $A$  by $\Omega$ is uncountable but every other section is countable.  Can anyone make me understand this theorem?  I can not understand what Mukres has tried to say  in his Topology's Book.. I can  not visualize what Mukresh has said in the proof.  I can not understand why we always get an element $\Omega$ such that the section less than $\Omega$ is countable. What ensures the availability of $\Omega$ in a uncountable well ordered set?  Can anyone make me understand with proper example?  Thank you in Advance.
Math people:  The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof.  It is not intuitively clear to me that uncountable ordinals or cardinals should exist.  What axioms of set theory are needed to accept their existence?  I think I have read here that the Axiom of Choice is not necessary.  I am reluctant to accept as "true" any theorems that rely on the Axiom of Choice.  EDIT: This question may be a duplicate: I just found  .  I apologize if this is so.
Will Windows 10's reset delete my Linux partitions? How to automount NTFS partitions? I have two NTFS partitions, and I don't want to mount them manually every-time I start Ubuntu.   How can I do this?  Is there a tool or a code to use?   If so, is it safe to automount? specially when they are being used by another OS? opengl storing multiple indices into indices buffer Most simple code to populate JTable from ResultSet I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again.  Consider the series:  $$1, \frac{1}{2},\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{3}{4}},\frac{\displaystyle\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{3}{4}}}{\displaystyle\frac{\displaystyle\frac{5}{6}}{\displaystyle\frac{7}{8}}},\cdots$$  Each fraction keeps its large bars while being put atop a similar structure. This can also be represented as $$\frac{1\cdot 4 \cdot 6 \cdot 7 \cdot\cdots}{2 \cdot 3 \cdot 5 \cdot 8 \cdot\cdots}$$ terminating at $2^n$ for some $n$, where it is much closer to the limit than elsewhere.  The challenge:   Find the limit, not too hard by experiment In the last expression, find a simple, nonrecursive, expression to say whether $n$ is in the numerator or denominator Prove the limit is correct-this is the hard one. Completely uninstall an app on Lollipop I am having a problem with uninstalling apps on Android 5.0 (Nexus 5). I am developing an Android app, so I debug/run the app directly on my device. For production, I exported the same app with my keystore, uninstalled the debug APK that I ran earlier, tried to install the signed APK, and I got this error:      An existing package by the same name with a conflicting signature is already installed.   Is it because the bytecodes are still in the ART runtime? Anyone has any idea how to solve this? My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number.  So we memorized this for future use in math competition.  But I don't understand why is it? Can you eradicate a disease without finding a Cure first? If you eradicate a disease in Pandemic without finding a cure first (ie remove all of one colour from the board) does it remain eradicated or can it still come back unless it has also been cured?  The rules are a bit unclear on this point ... Or I missed the rule that explicitly states this. There are a lot of software in Windows to merge PDF files but how can we do the same in Ubuntu? Infinite Product $\prod\limits_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$ Baking lots of bread - keeping a yeast starter The most common way to refer to non-immediate family? This is the PATH variable without sudo:  $ echo 'echo $PATH' | sh  /opt/local/ruby/bin:/usr/bin:/bin   This is the PATH variable with sudo:  $ echo 'echo $PATH' | sudo sh /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/X11R6/bin   As far as I can tell, sudo is supposed to leave PATH untouched. What's going on? How do I change this? (This is on Ubuntu 8.04).  UPDATE: as far as I can see, none of the scripts started as root change PATH in any way.  From man sudo:      To prevent command spoofing, sudo   checks ``.'' and ``'' (both denoting   current directory) last when searching   for a command in the user's PATH (if   one or both are in the PATH). Note,   however, that the actual PATH   environment variable is not modified   and is passed unchanged to the program   that sudo executes. Copy files from a list to a folder Changing user preferences via console? The machine that I'm currently using is an extremely slow netbook. Whenever I go to change user preferences in Blender, it stalls out. What I need to do is activate the "emulate 3 button mouse" checkbox. Is there a way to do this via console/script? How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. There's no need to  the  that have led to a lack of trust between  of  and . Suffice it to say that astoundingly they continue to assert that  and, instead of apologizing and reinstating Monica have demanded that she go through a process with no transparency, due process, or involvement from the community.   Has the time has come for Stack Exchange, Inc to spin off the community and all the data needed to keep the sites running to a separate non-profit, who would then in turn license the software from the company?  What's in it for the community?   The community can practice self determination for issues from as basic as the number of moderators they have to as fundamental as the policies they enforce. If the company development continues to be unresponsive to the community's needs, the community would be able to develop its own open source software that would meet its needs. They can find their own funding model instead of having ads imposed on them.   What's in it for the company?   Community actions will no longer reflect on the company. Interaction with the community as a customer may come more naturally to current leadership than the present paradigm. They don't have to worry about paying for the community's operating expenses. They can focus on doing one thing well. If $A=AB-BA$, is $A$ nilpotent? Let matrix $A_{n\times n}$, be such that there exists a matrix $B$ for which $$AB-BA=A$$  Prove or disprove that there exists $m\in \mathbb N^{+}$such  $$A^m=0,$$  I know $$tr(A)=tr(AB)-tr(BA)=0$$ then I can't.Thank you Methane’s absorption bands are centered at 3.2 and 7.2 microns -- far off the peak of the Planck spectrum for a 290 K blackbody near 16 microns.  Moreover, its absorption bands overlap with the water absorption region.  So why is methane considered a stronger greenhouse gas than carbon dioxide, which has an absorption band at 15 microns? Find count of all combination of numbers whose sum is x I want to find the sum of all combination of numbers whose sum is x, for e.g.  when x = 3 f(x) = countOf(111,12,21,3) = 4