I am trying to solve exercises from Pugh's Real Mathematical Analysis 2nd ed. and there is a question related to the connectedness of a disc. The question statement is as follows:  Is the disc minus a countable set of points connected? Path-connected? What about the sphere or the torus instead of the disc?  Intuitively, I think the disc is connected and also path-connected after removing those points. However, I would like to see proof of this. Can someone help me with that?
Here's a question that I share:  Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D $ is arcwise connected.
Why are continuous functions the "right" morphisms between topological spaces? Using Newton's Law of Universal Gravitation to find distance, with the question not giving sufficient information to substitute into the equation. Problem:  $F = \frac{(G m_1 m_2)}{r^2}$  An object of unknown mass is placed directly between the Earth and the Moon such that the forces of gravity acting on the object are equal from both the Earth and the Moon. The distance from the Moon to the object is exactly one-tenth of the distance from the Moon to Earth. Using the above formula, determine the distance from the Moon to the Earth.  Given is:  The factor $G≈6.67∙10−11\mathbf{N(\frac{m^2}{kg^2})}$ is a universal constant.  Mass of Earth $≈5.972∙10^{24}\mathbf{kg}$   Mass of the Moon as $≈7.348∙10^{22}\mathbf{kg}$  Mass of unknown object = $m$  My solution:  $F = \frac{(6.67 *10^-11) m (7.348*10^{24})}{R^2}$  $F = \frac{(6.67 *10^-11) (5.92* 10^22) m}{10R^2}$  As gravitational force acting on the object is equal   $\frac{(5.92\cdot 10^22)}{100r^2} = \frac{(7.348*10^{24})}{r^2}$  $R = 1.24 \cdot 10^4 KM$  Is this a viable solution to the problem? Please provide feedback. Thank you! Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know. Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$ NSDictionary with ordered keys I have an NSDictionary (stored in a plist) that I'm basically using as an associative array (strings as keys and values).  I want to use the array of keys as part of my application, but I'd like them to be in a specific order (not really an order that I can write an algorithm to sort them into).  I could always store a separate array of the keys, but that seems kind of kludgey because I'd always have to update the keys of the dictionary as well as the values of the array, and make sure they always correspond.  Currently I just use [myDictionary allKeys], but obviously this returns them in an arbitrary, non-guaranteed order.  Is there a data structure in Objective-C that I'm missing?  Does anyone have any suggestions on how to more elegantly do this? How can I search for a keyword with special characters in Google Search? I want to search for what += means, but Google won't return any documents.  Also when I come across some PHP code I don't understand I can't search for the keywords as written.  How can I search for these special characters or include these special characters in a search. Inspired by a , here's a list of the 'etymology' for the  hats. For pictures and a list of criteria, see  instead.  The list is a community wiki; feel free to edit it if you have more information about a hat. If you like to do some research, most hats have been featured in earlier editions, so you might be able to find more information in old Meta Stack Exchange questions. Salesforce - user trigger not invoked when user role changes There are two ways to change user role. One is from user detail page, which invoke user trigger. I am facing issue where if user role is changed from ROLE hierarchy screen, then user trigger is not being invoked.  Any body know if there is written documentation which specifies this? or is it just my ORG having some setting to skip user trigger?   Below is screenshot of the screen from where it happens. Are the square roots of all non-perfect squares irrational? Associativity of symmetric difference of sets By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.) How to remove the URL from the printing page? I want to remove the URL that gets printed on the bottom of the page.   like:  yomari.com/.../main.php?sen_n   How can it be omitted or prevent from getting printed?  To be more specific, is there any way I can prevent the page URL, date and the page title being printed along, while printing the web page? Volume of the torus with an integral Is there a rule to understand how the group "Cha" has to be pronounced?  "Character" sounds with a hard first syllable, while "Charm" sound softer, but I don't find how to tell which sound to use before earing someone saying the word. It could be because of the double consonant "rm" vs "ra"? Or is it just a matter of knowing the rule for every single word?  EDIT: More specifically, let's talk about UK English pronunciation. Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when is it the case?) that $\mathfrak{p}_1 + \mathfrak{p}_2 = K$ ?  I ask because I'm working through some lecture notes which frequently take some $K$ complete with respect to a discrete valuation, a prime ideal in a valuation ring  $\mathfrak{p} \subset \mathcal{O}_K$, a finite extension $L/K$, and then use the fact $\mathfrak{p}\mathcal{O}_L$ factors uniquely as some $\mathcal{P}_1^{a_1} \ldots \mathcal{P}_r^{a_r}$ (where $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$). Then we say $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$ $(\forall$ $ i \neq j)$.   However it isn't clear to me, because we've made so many different assumptions along the way, what the reason is that $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$. Is it because these primes occur in the factorisation of $\mathfrak{p}$, is it because of some property of Dedekind domains or fields, or is it not really possible to say out of context? Many thanks, and if anything needs clarification please ask. Calculating geometry for feature class with x/y and Lat/Long values in different projections? How do I change which audio jacks are used for input and output? I'm using a Realtek HD audio card built-in my motherboard. The Windows driver comes with a control panel that allows me to select which back panel jacks are used for what. So for example I can make both the blue jack and green jack for output and only the red one for mic-in. (Whereas by default, the blue jack is for line in, which I never need.)  How can I do the same under Linux? If possible, please don't suggest something that involves PulseAudio or JACK; I'd like to do it the plain way, e.g. by editing ALSA configuration files, if possible. The way I understand it, my problem should have nothing to do with software servers redirecting streams, just instructing the driver to treat this jack as so and so because it's hardware supported.  Thank you very much! $A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected Let $X$ be a topological space. Let $A$ and $B$ be two subsets of $X$ such that: $A$ and $B$ are closed, $A \cup B$ is connected, $A \cap B$ is connected     Prove that $A$ and $B$ are connected.   Thoughts:  Suppose that $A$ is disconnected. Since $A$ is closed, there are two non-empty, disjoint, closed sets $E$ and $F$ in $X$ such that $A = E \cup F$. Then I attempted many things but all were not really useful.  Please help. Thanks. Does there exist a holomorphic function with the following property? True or false: {{∅}} ⊂ {∅,{∅}} Is re-projection needed from SRID 4326 (WGS 84) to SRID 4269 (NAD 83)? I have US census data against state boundaries stored as  SRID 4269, in an MSSQL DB.  The input data used for calculations against that dataset is stored as SRID 4326.  As far as I can infer from the , SRID 4269 is just a subset of SRID 4326 and no coordinate re-projection is necessary.  Do I actually need to re-project from SRID 4326 to SRID 4269, provided that the input data is always within the bounds of the SRID 4269?