We can define the cofinite topology as the following $\mathcal{T}_{cofinite} = \{U \subseteq X| U = \emptyset, X-U  \mbox{is finite}\}$. Let $\mathcal{U}= \{U_{\alpha}\}_{\alpha \in I}$ be an open cover for $\mathbb{R}$. But where can we take this argument. We want to find a finite sub cover for $\mathbb{R}$ and then we are done. We can also included that since $\mathcal{U}= \{U_{\alpha}\}_{\alpha \in I}$ is an open cover for $\mathbb{R}$ that $\mathbb{R} = \cup_{\alpha \in I} U_{\alpha}$.
While we know that every finite set is compact, how to prove that every infinite subset of $\mathbf{R}$, the set of all real numbers, is also compact in the finite-complement topology (also called the co-finite topology)?   What is the situation in the case of an arbitrary infinite set $X$ with the finite-complement topology?
"Disappear" as a transitive verb Dark Mode for Stack Exchange sites Each Stack Exchange site should provide users the option of using a dark mode design/theme.  Why is this feature missing?  The major IDEs are already a letdown regarding dark color schemes. General relativity (gravitation) in time and one spatial dimension I don't have any idea of general relativity but intend to learn. Is it a good idea to study general relativity in two dimensions (time and single spatial dimension) in the begining to get good idea on the subject? If it is, then please give some references for such a treatment. Do not give the impression that all votes are unanimously in favor of the winning close-reason. I closed for "belongs on superuser," yet SO claims I closed for a completely different reason.  Source: How do I get the current location of an iframe? How do I resolve this installation error? I am getting the following error while installing Drush with composer require drush/drush.     Using version ^8.1 for drush/drush   ./composer.json has been updated   Loading composer repositories with package information   Updating dependencies (including require-dev)   Your requirements could not be resolved to an installable set of packages.            Problem 1   - drush/drush 8.1.0 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.1 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.2 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.3 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.4 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.5 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.6 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.7 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.8 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.9 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: don't install drush/drush 8.1.12   - Conclusion: don't install drush/drush 8.1.11   - Conclusion: don't install drush/drush 8.1.10   - Conclusion: don't install drush/drush 8.x-dev   - Installation request for drupal/drupal [No version set (parsed as 1.0.0)] -> satisfiable by drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: remove phpdocumentor/reflection-docblock 3.1.1   - Installation request for drush/drush ^8.1 -> satisfiable by drush/drush [8.1.0, 8.1.1, 8.1.10, 8.1.11, 8.1.12, 8.1.2, 8.1.3,   8.1.4, 8.1.5, 8.1.6, 8.1.7, 8.1.8, 8.1.9, 8.8.x-dev, 8.x-dev].   - Conclusion: don't install phpdocumentor/reflection-docblock 3.1.1   - drush/drush 8.8.x-dev requires phpdocumentor/reflection-docblock ^2.0 -> satisfiable by phpdocumentor/reflection-docblock [2.0.0,   2.0.0a1, 2.0.0a2, 2.0.0a3, 2.0.1, 2.0.2, 2.0.3, 2.0.4, 2.0.5].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.4, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.5, 3.1.1].   - Installation request for phpdocumentor/reflection-docblock (locked at 3.1.1) -> satisfiable by phpdocumentor/reflection-docblock [3.1.1].      Installation failed, reverting ./composer.json to its original   content. Why is $2\pi i \neq 0?$ How to change Gnome-Terminal title? Install CentOS 7 on HP dl120 gen9 server with b140i raid controller I'm trying to install CentOS 7 on a HP DL120 gen 9 server with a b140i controller. I have 4 1TB HDD setup in a raid 1+0 and the CentOS installer doesn't seem to recognize the logical drive created from the storage utility.  From my understanding i need some drivers for CentOS 7 to recognize the B140i controller. Also, the driver seems to be closed source by HP.  What is my approach here? Is there any other free linux OS that i can use on this server and that would recognize the raid controller?  Thank you. Subgroups of finite abelian groups. For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example.  I am aware of isomorphism theorems and classification of abelian groups, direct products etc. SSH and home directory permissions It took me hours to solve this SSH problem with one of my class accounts on my school's servers.    I couldn't ssh into one particular class account without entering my password, while passwordless authentication worked with my other class accounts. The .ssh/ directory and all of its contents had the same, correct permissions as the other class accounts.   Turns out the problem was the permissions set on my own home directory. Passwordless authentication did not work when the permissions on my HOME directory were set to 770 (regardless of the permissions set for .ssh/), but it worked with permissions set to 755 or 700.   Anyone know why SSH does this? Is it because the home directory permissions are too permissive? Why does SSH refuse to authenticate with the public/private keys when the home directory is set more permissive than 700? Is it: My apples and orange are/is wrong? Simple question:     My apples and orange are wrong   or     My apples and orange is wrong   I am not a native English speaker, and I am having some trouble choosing between plural are or singular is for that kind of example. I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos\left(\frac{\pi m}{2}\right)\Gamma (m+1)$$  for $-2&lt; Re(m)&lt; 0$ What finite groups always have a square root for each element? Natural log approximation I've got an equation that contains   $$x^p - 1$$   $x$ is any positive number (such as 2) and $p$ is a small positive number close to 0 (such as 0.001).  For some reason (that I may have known in High School), this can be approximated by   $$\ln(x) \times p$$  Can anyone explain why this approximation works?  Does it have something to do with approximating a natural log with a series? Can we get x comment votes per y minutes, rather than one vote per 5 seconds? transpose columns to rows It isn't any of the words on this  and I believe it started with the letter 'e' (though please offer your word even if it doesn't).  Example sentence: Because he researched what he had heard from his friends, he was able to refute many of their _________________.  Factoid seems to be the closest one, but the word I am looking for only refers to a belief that is false. Two "offtopic" flag reasons The Earth in all of its varieties has wide ranging climates. At the top level it has tropical, dry, temperate, continental, and polar climates. And yet often the planets depicted in science fiction are often single climate planets.    The forest moon of Endor The desert planet of Dune The desert planet of Tatooine The desert planet of Vulcan The outer planets in Firefly all appeared to be mostly desert (supiciously like Calfornia desert).  The inner planets in Firefly were all temperate or tropical.  The icy Breen homeworld The icy Andorian homeworld The icy Frost Giant planet   I'm not suggesting there are no Earth-like planets in science fiction, but it feels like it is less common than it should be. Considering the only planet we have as an example is our own, it seems like planets should be just as varied in the cosmos.