How do I prevent mails sent through PHP mail() from going to spam?
How do you make sure email you send programmatically is not automatically marked as spam?
When NASA sends astronauts to space, they have to wear uncomfortable suits. Why aren't the humans in Star Wars spacecraft doing the same thing? What technology do they have on board that remove this need? How can I search for a keyword with special characters in Google Search? No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof? How to give a direct proof of the following result?     Let $A$ be a subset of $\mathbb{R}$ such that $A$ is both open and closed. Then $A$ is either empty or all of $\mathbb{R}$?   My work:   If $A$ is not empty, let $u \in A$. Then there is some open interval $(a,b)$ such that $$u \in (a,b) \subset A.$$  Now if $\mathbb{R} - A$ is not empty either, then let $v \in \mathbb{R} - A$. Then there is some open interval $(c,d)$ such that  $$v \in (c,d) \subset \mathbb{R} - A.$$  Suppose that $u &lt; v$.   Since $$\emptyset \subset (a,b) \cap (c,d) \subset A \cap (\mathbb{R} - A) = \emptyset,$$ we must have $$(a,b) \cap (c,d) = \emptyset.$$ So we can conlclude that $$b \leq c.$$ But $b \in A$ and $c \in \mathbb{R} - A$. So we must have $$b &lt; c.$$  What next? Can anybody here please help complete the proof from here on?  An edit based on a comment by Marc Paul:   Let us assume that the set $A$ is a non-trivial clopen set in $\mathbb{R}$. Let us define a function $f \colon \mathbb{R} \to \mathbb{R}$ as follows:  $$f(x) \colon= \begin{cases} 1 \ \mbox{ for } \ x \in A; \\ 0 \ \mbox{ for } \ x \in \mathbb{R} - A. \end{cases}$$   Let $V$ be an open set in the range space $\mathbb{R}$. We show that the inverse image set $f^{-1}(V)$ is open in the domain space $\mathbb{R}$.   The following cases arise:   If $0, 1 \in V$, then $f^{-1}(V) = \mathbb{R}$.   If $0 \in V$ but $1 \not\in V$, then $f^{-1}(V) = \mathbb{R} - A$.   If $1 \in V$ but $0 \not\in V$, then $f^{-1}(V) = A$.   And, if $0 \not\in V$ and $1 \not\in V$, then $f^{-1}(V) = \emptyset$.   Thus, $f^{-1}(V)$ is open in the domain space $\mathbb{R}$.   Hence the function $f$ is continuous.   What next? How does this lead to our desired conclusion?   [Yet another edit, again based on valuable comments from Marc Paul: ]  So if both $A$ and $\mathbb{R} - A$ were non-empty, then let's suppose $a \in A$ and $b \in \mathbb{R} - A$, and we can assume without any loss of generality that $a &lt; b$.   Then $f(a) = 1$ and $f(b) = 0$. So by the intermediate value theorem there is a real number $c \in (a,b)$ such that $f(c) = 1/2$, which is a contradiction because the image set of $f$ does not contain $1/2$. Java: split a List into two sub-Lists? Change web folders names with uppercase to lowercase or not? I have  today and one member was voted as duplicate and private feedback was there, I choose no and submitted the form. When I revisit the question or refresh the page same notice reappears with yes/no options.  In-network tab I got 200 OK status and response as "success":true  Environment: OS: Windows 10 Browser: Chrome 79.0.3945.117 (Official Build) (64-bit)  From the comments, the same issue is happing with   (might be with whole StackExchange network)      From network tab Terms for collections of animals As I watched the murder of crows sitting on the line above my house this evening, I got wondering where all of the collective nouns for animals (pod of whales, gaggle of geese, pride of lions) came from and why we need so many.    If sheep can be a flock, why can't whales, geese, lions, and crows? What do people who are using a pensieve look like? As far as I remember from the books, there's no description of what a person using a Pensieve looks like. When Harry is using the Pensieve, he describes it as if he were falling into it, but it's never stated whether his body really enters the Pensieve, because the experience is just told from his point of view.   Now let's say Dumbledore enters his office while Harry was using the Pensieve -- what would Dumbledore see? Is there an answer in the books/canon? How can I calculate sensor size (width and height) from focus length, vertical, and horizontal angle of view? How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. How to model non-negative zero-inflated continuous data? I'm currently trying to apply a linear model (family = gaussian) to an indicator of biodiversity that cannot take values lower than zero, is zero-inflated and is continuous. Values range from 0 to a little over 0.25. As a consequence, there is quite an obvious pattern in the residuals of the model that I haven't managed to get rid of:   Does anyone have any ideas on how to solve this? What is the mechanism behind the slowdown of light/photons in a transparent medium? Remove space around tikz How can I remove ALL space around TiKZ. So given MWE below, I want no space between fbox and the node box (so essentially I want to see 1 line). How to realize this?   Is this space in tikzpicture or in \fbox?   How to remove space between \fbox and node    MWE  \documentclass{article} \usepackage{tikz}  \begin{document} \fbox{%   \begin{tikzpicture}[node distance = 0.8cm and 0.2cm]%     \node[draw] (center) {CENTER}; %   \end{tikzpicture}% } \end{document} I would like to determine the operating system of the host that my Java program is running programmatically (for example: I would like to be able to load different properties based on whether I am on a Windows or Unix platform). What is the safest way to do this with 100% reliability? What do you do when the bus bike rack is full? Sometimes I take commute via bike, other times on the bus, but I rarely plan on doing both because I worry about the possibility of waiting for a bus only to find that its bike rack is full and having to wait for the next bus, which may also have a full bike rack.  What do you do in this case? I asked a bus driver today on the way home and I couldn't really understand what he was saying, but he said something about a bike rack in the back and I think he was saying that were he not driving the rapid bus and were his schedule different, he would put any extra bikes in the back. Not sure though, he could have said something completely different. Is there a built-in function to print all the current properties and values of an object? So what I'm looking for here is something like PHP's  function.  This is so I can debug my scripts by seeing what's the state of the object in question. Resetting theorem counters: \section and (missing) \subsection Adding HTML entities using CSS content How do you use the CSS content property to add HTML entities?  Using something like this just prints &amp;nbsp; to the screen instead of the non-breaking space:  .breadcrumbs a:before {     content: '&amp;nbsp;'; } Is Lambert Academic Publishing a reputable company? I was contacted by the Acquisition editor of Lambert Academic Publishing (LAP) to publish my bachelor's thesis as a printed book. Is LAP any good? I have no idea about it; I have just heard that it's a freelance publishing house. Is it true that  $$\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$$  Just making sure.   If not, how about:  $$\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$$   I'm having a hard time concentrating right now, and I'm trying to figure out how to get the area under a curve when the function is inverted.