Suppose that $C$ is the cantor set and let $\sim$ be an equivalence relation on $C$. Are there necessary and sufficient conditions on $\sim$ to guarantee that the quotient $C/\sim$ is Hausdorff?  What I know:   If $q: X \to Y$ is an arbitrary quotient map then $Y$ is $T_1$ if and only if the fibres of $q$ are closed. So the equivalence classes of $\sim$ must be closed in $C$. Since $C$ is compact and normal, the results listed in   imply that if the quotient map $q: C \to C/\sim$ is closed then $C/\sim$ will be Hausdorff. What conditions can be put on $\sim$ to guarantee that the quotient map is closed?
$X$ is a Hausdorff space and $\sim$ is an equivalence relation.  If the quotient map is open, then $X/{\sim}$ is a Hausdorff space if and only if $\sim$ is a closed subset of the product space $X \times X$.  Necessity is obvious, but I don't know how to prove the other side. That is, $\sim$ is a closed subset of the product space $X \times X$ $\Rightarrow$ $X/{\sim}$ is a Hausdorff space. Any advices and comments will be appreciated.
It's been said that .  It the corresponding statement true for typical cryptographic hashes, like MD5, SHA-1, SHA-2 or SHA-3?  For example, would SHA-512 be bijective when hashing a single 512 bit block? What is the difference between "likelihood" and "probability"? I've found in Wikipedia the formula for calculating an individual row in Pascal's Triangle: $$v_c = v_{c-1}\left(\frac{r-c}{c}\right).$$ where $r = \mathrm{row}+1$, $c$ is the column starting from $0$ on left and $v_0 = 1$.  Now, I've tried to do by hand and it works,  but I don't understand how to find this magic formula when I don't have access to Wikipedia...:) How can I refinance or get out of a car loan I am upside down on? I purchased a car a little over 1 year ago a 2012 Dodge Avenger and recently went to get it refinanced and found out that because I'm so upside down that I can't get it refinanced I still owe over 15,000 and the car is only worth about 8000. I'm so bothered by this a few days ago I received a pre-approval letter from capital one bank for 35,000 so how can I go about doing this if I trade the car in I will have to put down a substantial amount of money in order to get out of this car. Can I just get the other car and turn the other one in. What should I do Winter 18  does not get values from a parent record Problem with MGI CRS on the Balkan Are there rules for creating level 0 characters? Are there rules for creating level 0 characters in 5e? I've been running a campaign where the players begin sub-standard and 1st Level characters are generated in-game as the story unfolds. I just made them all begin with a couple of hp and 9's across the board for ability scores (since I wanted them at a disadvantage compared to the average human, and to fail at a generic DC10 task more often than succeed).  It was a real grind, with many opportunities for TPK (total party kill) when faced with even a single challenge rating 1/2 adversary (obviously I fudged the dice a little to prevent this). Are there published rules for this?  If there aren’t published rules for this, has anyone tried doing something like this? Any first hand experience on how it has played out would be welcome. pattern matching in vim Unknown Column In Where Clause It's said that if  is spared in  quest, he remains in the Dawnstar sanctuary and can be later hired as a follower. Well I did just that and I can't find him anywhere. Where exactly is he? Does he disappear and only reappear after some other moment in the game? A fleet of nine taxis is dispatched to three airports, in such a way that three go to airport A, five go to airport B and one goes to airport C.  If exactly three taxis are in need of repair. What is the probability that every airport receives one of the taxis requiring repairs.  My method was total number of ways is (9!)/(3!*5!*1!)= 504 (this is correct)  Now you have three groups and of every group one position is already filled. So: ....R, ..R, R So the numbers of ways to distribute the rest is: (6!)/(2!*4!)=15  But according to answer model of my professor, it should be 90. So I assume he says the number of ways of distrubiting these 3 broken cabs over the 3 spots is 3*2*1. But to me this seems wrong? The broken taxis don't differ so R(1)R(2)R(3) is same as R(2)R(1)R(3). If you standing at the airport A, it doesn't matter which of the broken cabs stands there? My installer build "signs" a DLL using a Code Signing certificate during the build process.   I've noticed that if I try to build twice in succession, the second build fails because the DLL is already signed so signcode chokes. Obviously I can fix this by signing a copy of the DLL in the build, but the problem intrigued me:   Is it possible to "unsign" a DLL, and if not, why not...? $ not defined using jQuery in WordPress I know that jQuery is loaded, because I can switch out the $ for 'jQuery' and everything behaves as expected, but this will be a messy script if I can't fix this  This script:  jQuery(document).ready(function(){     $("ul.vimeo_desc_feed li a").click(function(){         alert($(this).attr('href'));         return false;     }) });   Produces the error $ is not a function  This script:  jQuery(document).ready(function(){     jQuery("ul.vimeo_desc_feed li a").click(function(){         alert(jQuery(this).attr('href'));         return false;     }) });   works fine. Is $\le$ defined by $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2 \land𝑦_1 \ge 𝑦_2$ a linear order? I need to tell if the relation $\le$ defined by relationship $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2$ $\land$ $𝑦_1 \ge 𝑦_2$ linear order?   I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help! Profitable strategy in coin tossing? There is a coin with a probability $p$ of heads, and $1-p$ of tails. Tosses are independent of each other. When you bet an amount of money $x$, you receive $2x$ if it lands heads, and you lose what you bet if it lands tails.  We are going to use the following strategy:  You start betting an initial amount $x_0$. Every time you lose, you bet twice what you bet in the previous throw. You keep doing this, until you win, at which point you retire, or start again, betting $x_0$.  The interesting case of course is when $p &lt; 0.5$.  The question is: How profitable is this strategy? Intuitively I feel that eventually I can always win an arbitrarily large amount of money. But that can't be entirely right ("the house always wins!"). So what am I missing here? What's the expected win in each run?  Also, does this sort of strategy have a name in the literature?  P.S.: Found this later, Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$.  I know that the number of elements of the power set of $A$ is $2^n$, i.e.  $|\mathcal{P}(A)|=2^n$.  I came across the fact that exactly half of the elements of $\mathcal{P}(A)$ contain an odd number of elements, and half of them an even number of elements.  Can someone prove this? Or hint at a proof? With reference to:    I removed the Cortana packages from my running Windows 10 using the method posted by magicandre1981 and using  win6x_registry_tweak.   My question is how can I re-install Cortana (or for that matter any other package removed similarly)?  I have opened up the install.wim image (converted from a Win 10 install.esd file) and found Cortana in the system apps folder, but don't know how / what to use to re-install it.  Any help please? .NET Casting Generic List In a LaTeX article, the math characters are more "curvy" than the text characters:  \documentclass[12pt]{article} \begin{document}  $x^2 + y^2 = z^2$ \end{document}     In a beamer presentation, the math characters do not have the same curves:  \documentclass{beamer} \begin{document} \begin{frame}  $x^2 + y^2 = z^2$ \end{frame} \end{document}     Question  How can I get the beamer math to look just like the article math?  I would prefer if the solution was a single line (that I could place at the top of my LaTeX file) as opposed to something that needs to appear next to every piece of math. I have a Mac that runs Snow Leopard and I intend to upgrade it to Mountain Lion and sell it on ebay. However, I only have one hard-drive, and I'm sure I can't use something in my OS to delete my OS and all user data.  Is there something for this that's built into the firmware that I can do on boot, or otherwise?   I have source code on my Mac for multiple corporations' proprietary software and I can't risk somebody gaining access through the use of any sort of disk recovery tool. I plan to use a 7 pass erase rather than simply deleting the directory.