If I have a quotient map $q : X \rightarrow Z$ and then I form a map $q\times \text{id}_Y : X\times Y \rightarrow Z\times Y$, apparently this is only a quotient map if $Y$ is locally compact and Hausdorff.  Why is this? What goes wrong if $Y$ isn't locally compact or Hausdorff? I would think that the openness of the preimage of an open set under this map only depends on the first coordinate since the second one is just the identity on $Y$.
I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff.
iOS navigation bar item image size Search using symbols is broken What is the difference between "citizen" and "denizen" Citizen: 1. A legally recognized subject ornational of a state or commonwealth, either native or naturalized. 2. An inhabitant of a particular town or city.  Denizen: 1. An inhabitant or occupant of a particular place.  Same thing? Tikz, Tree without overlaps I'm trying to get a nice tree without the overlaps, can anyone help me to improve my code?  \documentclass{standalone} \usepackage{amssymb} \usepackage{amsmath} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{dsfont} \usepackage{tikz} \usetikzlibrary{shapes} \usetikzlibrary{graphs}  \begin{document}       \begin{tikzpicture}[baseline,level/.style={level distance=20mm, sibling distance =35mm/#1},scale=1]         \node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                 child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         }                     }                 }                    child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                 }             };     \end{tikzpicture} \end{document} I saw a code change at work, where the mode values were changed from 777 to 0777 to make nfs setattr work. What is the difference in the 2 values? 1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:  if $C = A+B$  or  $C = A-B$ $$\Delta C = \Delta A + \Delta B$$ if $C = AB$ or $C = \frac{A}{B}$ $$\frac{\Delta C}{C} = \frac{\Delta A}{A}  + \frac{\Delta B}{B}$$ if $C = A^m$ $$\frac{\Delta C}{C} = |m|\frac{\Delta A}{A}$$   These formulas are derived from the expression for the differential of a function of multiple variables:  if $C = f(A,B)$ $$ dC = \frac{\partial C}{\partial A}dA + \frac{\partial C}{\partial B}dB \Rightarrow \Delta C = \frac{\partial C}{\partial A}\Delta A + \frac{\partial C}{\partial B}\Delta B$$  and I think that makes sense because the value C is just a function of two variables that happen to be measurements.  2) But this morning, I was told that this is wrong and I should actually use these instead:  if $C = A+B$  or  $C = A-B$ $$(\Delta C)^2 = (\Delta A)^2 + (\Delta B)^2$$ if $C = AB$ or $C = \frac{A}{B}$ $$\left(\frac{\Delta C}{C}\right)^2 = \left(\frac{\Delta A}{A}\right)^2  + \left(\frac{\Delta B}{B}\right)^2$$  which are apparently derived from a general formula:  $$(\Delta C)^2  = \left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...$$  or   $$\Delta C  = \sqrt{\left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...}$$  3) So:   Why is this formula better?  Where does it come from? What does it actually represent? (Do I recognize the shape of a norm in that last formula?)  What is wrong with the formula of the differential?   I'm asking a lot, but the propagation of uncertainty always confused the ship out of me. Determine destination location of apt-get install ? Why was the switch statement designed to need a break? Given a simple switch statement  switch (int) {     case 1 :     {         printf("1\n");         break;     }      case 2 :      {         printf("2\n");     }      case 3 :      {         printf("3\n");     } }   The absence of a break statement in case 2, implies that execution will continue inside the code for case 3. This is not an accident; it was designed that way. Why was this decisions made? What benefit does this provide vs. having an automatic break semantic for the blocks? What was the rationale? Can't resolve issue: iptables: No chain/target/match by that name For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^TAX)]$? Please note that $A$ is not a random variable. Making crafted items usable in adventure mode In a map I'm making the player has to survive while locked in a maze. To stop them from breaking out, they will be in adventure mode. However, I still want them to be able to craft items and use them with specified blocks. I have tried several methods already but they all have a flaw.  One method is to have a scoreboard objective called item and giving certain item entities a specific score using something like:  /scoreboard players set @e[type=Item] item 1 {Item:{id:"minecraft:sapling"}}   Then it is possible to select these items with:  /entitydata @e[type=Item,score_item=1,score_item_min=1] {Item:{tag:{CanPlaceOn:["dirt"],HideFlags:24}}}   This works but it requires the item to be dropped, which isn't a problem for saplings, but for tools and other craftable items, it becomes a hassle.  To fix this, I'm currently clearing tools without damage and giving the player a slightly damaged tool in return:  /clear @a minecraft:stone_axe 0 1   /give @a minecraft:stone_axe 1 1 {CanDestroy:["log","log2","leaves","leaves2"],HideFlags:24}   This doesn't work for items without damage values (e.g. torches), but I don't mind these having to be dropped as there aren't too many of these. However, there is another problem. In the nether, the player will have access to an enchantment table and an anvil. This means they can repair enchanted tools, which will cause them to be cleared and replaced with unenchanted ones as soon as they go back to the overworld.  I also thought about using a stat.craftItem scoreboard objective and giving them the usable item if they craft it. However, this would result in the player having a duplicate of every item they crafted, as I think it is impossible to select an item without NBT-data in the /clear command.  Is there any way to make this work? And if not, would it be better to have the player drop all crafted items or have duplicates of them all? Apache: Scammers point dozens of domain names at my dedicated IP. How to prevent this? For the sake of discussion, pretend that I have an Apache webserver which serves a website at www.example.org. This domain name resolves to the public IP address at 192.168.1.100 (pretend this is a public IP).  The discussion at  demonstrates that anyone can point their domain at my IP. Apache won't prevent this by default, but an administrator can deal with these one at a time through use of a blacklist.  However, it's becoming more common for scammers to register dozens of domains (or more) for use in a scam. I'm concerned about the following possible attack:   A scammer registers hundreds of domain names, and points them at my dedicated IP address. The scammer directs search engines at their scam domains, thus using my content to increase their standing in the search engines. Later, the scammer moves their domain names to point to their own servers which host a scam/porn, or a competing business, etc. thereby reaping the benefits of their standing in the search engines. Some of these domains might be used to in  as well. Profit!!   I believe I have seen scammers using this trick in the past, involving dozens of domains. We didn't realize the scam implications at the time, and assumed these were misconfigured domains.  Is there a term for this nefarious SEO trick? SEO masquerading? DNS Diversion?  How could I prevent this using Apache? I'm considering a "whitelist" fix based on using a default VirtualHosts, ServerNames &amp; ServerAliases so that Apache only responds to requests where these white-listed ServerNames appear in the "Host: " header. Everything else would be rejected (or redirected to a particular page). However, I am unsure if this is the best approach.  For example, I've configured the domain  to point at the IP for Serverfault.com. You can see the results here:  . Why doesn't the UK ask for travel bookings for a tourist visitor visa? New profile pages are out - bugs and feedback master list Update:  Thanks for all the feedback and bug reports! I am locking this thread so people don't keep answering it when we no longer monitor it.  If you see a bug or want to raise a feature request, please do so in the usual manner, as a new question.    Some of you may have noticed that  the  and  pages that we have been talking about for a while now on .  Most? Well, yes, the new design is dependent on a SE network-wide update (a new base css framework) that the design team have been working on. This is an ongoing effort and we will be launching the new pages as sites are moved to the new design.  You've all been hugely helpful already - much of what we love most in here came from  and  you gave us on Meta.     But it's possible that it's not quite perfect yet. If you find any bugs and have any feedback to give, please post these as answers to this post.  Just post one issue per answer - that would help us focus on specific items one at a time. Let $X$ and $Y$ be finite sets. Then the set $Y^X$ is finite and $\#(Y^X) = (\#Y)^{\#X}$.  I can see how to do it combinatorially. Let $y \in Y$. There are $\#(X)$ choices for such y to come from x, therefore we have $\#Y^{\#X}$ choices. I am not sure how to do it bijectively. How many named characters died in Avengers: Infinity War? How many named characters have died       and/or disintegrated      by the end of Avengers: Infinity War. It is difficult to speculate who died on         Xandar and the Asgardian ship.       So restrict it to on-screen deaths only. I'm very confused about tenses.  I have examples in two different situations here. Situation 1: I went to New York two months ago and was talking to someone about it.  A. This is the first time I've been to New York. B. That was the first time I'd been to New York.  Which is correct in a daily conversation?  Situation 2: I'm going to New York next month and talking to someone.  A. This will be the first time I've been to New York. B. This is the first time I've been to New York. C. This will be the first time I go to New York.  Which sentence is appropriate?  According to a grammar article I've read before, I think it says that I can use this sentence in both cases:  This is the first time I've been to New York.  But I'm not sure if this is really so. So I'd like to know which tense I should use in each case. Could you explain it? How do I add polygons to fill a face? trying to recreate this Elijah bottle. I have two sides, one of which is outlined in red. How do I fill that face with polygons so I can then grab the center (that red square dot) to pull it outward thereby creating the slight outward bowing of the bottle? How do I create an Excel (.XLS and .XLSX) file in C# without installing Microsoft Office? How can I create an Excel spreadsheet with C# without requiring Excel to be installed on the machine that's running the code? I've seen a lot of people talking about FPGA's before and I know that it stands for field-programmable gate array but how does it work and what is the purpose of using an FPGA?