I have the following question on a worksheet: Consider the finite complement topology τf on N. Does this topology have a countable base? If so give one such base and if not prove your claim.  Been trying to solve this for several hours but I have got nowhere. I think that there is no countable basis because it seems like to construct a basis you would need to take the entire topology, but I have no idea if that's right or how to show that.   Thanks
I am working on the problems from the textbook "Topology without tears". I am stuck with problem number $4$ in Exercise $2.2$. Could anyone suggest some hints on how to proceed? The question goes as follows.  A topological space $(X,\tau)$ is said to satisfy the second axiom of countability or to be second countable if there exists a basis $\mathcal{B}$ for $\tau$, where $\mathcal{B}$ consists of only a countable number of sets.   Prove that the discrete topology on an uncountable set does not satisfy the second axiom of countability. Let $(X,\tau)$ be the set of all integers with the finite-closed topology. Does the space $(X,\tau)$ satisfy the second axiom of countability?   For the first problem, I tried to argue by contradiction assuming that the basis for the topology is countable. But I did-not know what I need to look at to prove the contradiction. For the second one, I do not know where to start. I have not thought deeply on the second one though.  Thanks, Adhvaitha
Product Notation for Multiplication in Reverse Order Is there a standard notation for multiplication in reverse order? For example consider the problem  $$x_{k+1} = A_k x_k$$  where $x_i \in \mathbb{R}^n$ and $A_i \in M_n(\mathbb{R})$, ($i=0,1,2,\dots$) without any further assumptions on $A_i$. The solution to this problem is  $$x_k = A_{k-1} \dots A_1 A_0 x_0$$  Obviously, writing  $$x_k = \prod_{i=0}^{k-1} A_i x_0$$  is wrong because of the multiplication order. But I'm also unconfortable to use  $$x_k = \prod_{i=k-1}^{0} A_i x_0$$  as the notation suggests that we need to increment $i$, not decrement it. Also, one may need to write $\prod_{i=k}^{m} A_i$. Then, we need to explicitly state $m &lt; k$ to imply ordering.  I can always write the open form but as similar products frequently occur in calculations in my work I want to express them in a more compact way. The question is, is there a standard and nice notation to do so? There are some suggestions in , but I think none of them are "nice enough" to work with.  I came up with the notation  $$x_k = \coprod_{i=k-1}^{0} A_i x_0$$  to imply decrementing $i$ rather than increment it, but I don't know if it is used in somewhere else or there is a standard notation. How to us possessive apostrophe with words in quotes? For example,   ...a few days later I discovered that those five little boys were not that well-behaved (as I firstly thought). In fact they were quite mischievous and conniving. The broken tree next to my house and the cat with its legs tied were all their "inventions". And Lilly, as it later turned out, was able to get to Denver so fast not without those "little inventors'" help. How did Suren Manvelyan create the macro pictures of eyes? I came across some really cool  today. How did he manage to get these photos? I couldn't find any EXIF information anywhere, and I don't have enough experience with macro to figure out how he did it.  It's obvious that the light has a lot to do with the result. How and what kind have been used? And what kind of camera equipment? I have a 24/7 always-on Debian Jessie based headless home server that has a large 1TB SSD for the OS and all of my frequently accessed files. This same system has 4 larger hard disk drives in a SnapRAID array. These are mainly for archiving infrequently accessed Blu-rays and want those drives to remain spun down in standby unless I actually read or write to them. They are all formatted as ext4 and mounted with noatime and nodiratime enabled.  So even though no process or program should be regularly accessing those drives in any direct way, the hard drives constantly get spun up from standby. It seems to be related to graphical programs that provide a gui file browser, even something like Chromium. If I don't even browse into those drives, I'm thinking that these processes by simply getting a list of available drives spins up the hard disks. Much like blkid does. The problem is, it's hard to determine the root cause of this since none of these processes are actually reading or writing the filesystem on those drives, so no files are actually changing or being touched. Is there some sort of cache that I can populate or a buffer to prevent these programs from spinning up the hard drive simply by getting a list of available disks? This is honestly driving me insane, since I can't find a reliably way to keep these disks spun-down even though there is no direct access of the filesystem.  UPDATE: Thanks to Stephen's answer, I was able to trace the disk activity to gvfs and udisks. It's a real shame that these processes insist on waking up disks in standby when they aren't actually being accessed to do any real I/O with the filesystem. So far I just uninstalled them, knowing that it will remove some functionality from PCManFM and the like. Create Random Number Sequence with No Repeats Duplicate:    I want an pseudo random number generator that can generate numbers with no repeats in a random order. For example: random(10) might return 5, 9, 1, 4, 2, 8, 3, 7, 6, 10 Is there a better way to do it other than making the range of numbers and shuffling them about, or checking the generated list for repeats?  Edit: Also I want it to be efficient in generating big numbers without the entire range.  Edit: I see everyone suggesting shuffle algorithms. But if I want to generate large random number (1024 byte+) then that method would take alot more memory than if I just used a regular RNG and inserted into a Set until it was a specified length, right? Is there no better mathematical algorithm for this. $3^{n+1}$ divides $2^{3^n}+1$ Describe all positive integers,n such that $3^{n+1}$divides $2^{3^n}+1$. I am little confused about what the question asks-if it asks me to find all such positive integers, or if it asks me to prove that for every positive integer n,$3^{n+1}$ divides $2^{3^n}+1$. Kindly clarify this doubt and if it's the former part, please verify my solution-n=1. Please will someone explain what time dilation really is and how it occurs? There are lots of questions and answers going into how to calculate time dilation, but none that give an intuitive feel for how it happens. How does a lurker gain reputation to receive privilege for commenting? I have been reading Stack Exchange for years - mostly not logged in, and have never posted (this is my first). I'm simply not very talkative, and very good at googling.  Recently a few occasions have come up when I've wanted to comment on a question.  I wanted either to ask for, or provide, clarification which did not warrant a new answer.  How do I gain enough reputation to comment without:   asking questions which have already been answered or that I can figure out on my own, or posting my comment like as an answer (i.e. being  rude)? Can a smaller sensor's "crop factor" be used to calculate the exact increase in depth of field? If APS-C and similar crop-sensor digital cameras have a focal length multiplying effect such that a 50mm lens has an apparent focal length closer to the field of view of an 80mm on a full frame camera, and yet at the same time the depth of field for the smaller sensor camera is more like the depth of field a 50mm lens would produce on a full frame camera (using the same aperture), then this would seem to suggest the concept of an "aperture dividing effect."  In other words, a 50mm f/1.8 lens on an APS-C camera would act more like a 80mm f/2.8 (approx. 1.8 * 1.6x) lens in 35mm equivalent — for depth of field, not considering exposure.  Can someone with a better understanding of the physics involved clarify this for me.  I've never seen this concept explicitly mentioned anywhere, so I am a bit suspect of it. Create a Collection inside another Collection with python How to create a Collection inside another existing Collection with Python? By looking at an integral and bounding the error? How many Shutter Clicks on a New Camera considered normal? How many Shutter Clicks on a New Camera considered normal?    I purchased a quotes new Nikon D750 and checked the shutter count and it has 264 Shutter Clik count on it.    Is that suspect of a returned purchase or used camera? What is /etc/mtab in Linux? What is /etc/mtab in Linux?   Why is it needed and advantages of having it? Why don't we add a question with the same tag on multiple Stack Exchange sites? Illogical GTK theme behaviour on Ubuntu 18.04 What does general relativity say about the relative velocities of objects that are far away from one another? What does general relativity say about the relative velocities of objects that are far away from one another? In particular:--  Can distant galaxies be moving away from us at speeds faster than $c$? Can cosmological redshifts be analyzed as Doppler shifts? Can I apply a Lorentz transformation in general relativity? Wrong hyperref with repeated counter I'm having an interesting problem with hyperref.  Apparently, when a counter is reset by \setcounter{mycounter}{0} then hyperref always links to the first possible instance of the counter and not the one where the respective label sits.   Check this MWE out:   % !TEX TS-program = xelatexmk \documentclass{article} \usepackage[paperwidth=6in,paperheight=3in]{geometry} \usepackage{hyperref} \hypersetup{colorlinks=true, allcolors=blue, } \newcounter{mycounter}\setcounter{mycounter}{0} \newcommand\countme{\refstepcounter{mycounter}\themycounter}  \begin{document}  First line \countme{} has some interesting text.  The second line \countme \label{onelabel} has \textbf{more} interesting text.  As we saw at line \ref{onelabel} ...  \newpage  A new page brings new numbers \setcounter{mycounter}{0}   A very different first line \countme{} has other interesting text.  The second line \countme \label{otherlabel} has \textbf{different} interesting text.  As we saw at line \ref{otherlabel} ... \end{document}   As you can see in the screenshot below, the pop up preview of the \ref on the second page doesn't refer to the second line with \label{otherlabel} but to the one on the first page (also has the tooltip "Go to page 1" -- in TeXShop).   Am I not understanding something fundamental about hyperref?      The context for all this is an environment that resets my counter to 0 at begin.  And all subsequent \refs to labels in a second or third iteration of the environment point to the very first instance of that environment, not the correct one where the label sits. Constants of integration in integration by parts I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right? How to find all the subclasses of a class given its name?