Can this set of subsets of $\mathbb{N}$ be uncountable? Let $S$ be a collection of subsets of $\mathbb{N}$ such that for every $A, B ∈ S$ we have $A ⊂ B$ or $B ⊂ A$. Can $S$ be uncountable?  I think we don't have to be concerned about elements of $S$ which are finite subsets of $\mathbb{N}$, as finite subsets can be listed by their size according to this property. But I am not sure how to treat infinite subsets.  It would be much appreciated if comments and answers come in the form of hints rather than complete solutions as I think I'd benefit more that way.
Finding an uncountable chain of subsets the integers How to find an uncountable subset of $P(\mathbb{N})$ such that every two elements of it can be compared. In fact, give an uncountable subset of $P(\mathbb{N})$ such that has totality property. We mean by $P(\mathbb{N})$, the powerset of natural numbers set $\mathbb{N}$.
Converting a system command to python for file search and delete ISP blocking port 25 Correct tracked camera orientation Anyone know a simple method to swap the background color of a webpage using JavaScript? Best way to recover mysql from a "chmod -R 777 /" with databases intact Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact This question is from Wayne Patty Exercise 4.2  Let $T$ and $U$ are Topologies on set X. Suppose $ (X,T) $ and $(X,U)$ are compact Haurdsorff Spaces. Prove that $T=U$ or $T$ is not a subset of $U$ or $U$ is not a subset of T.  I thought of proving it by this way: Assuming that if $T$ is a subset of $U$ or $U$ is a subset of $ T$ then $T=U$ must be true. But I was unable to think on how to use compactness or hausdorff property and was unable to proceed. Well, also this question was asked before on this website. But I am unable to understand how the solution will work despite trying to understand it for a fair amount of time as it was just outlined. So, I am looking for a detailed solution. I came across the following problem in a book:  Give a combinatorial proof of  $$ {n \choose 0} + {n \choose 2} + {n \choose 4} + \, \, ... \, = {n \choose 1} + {n \choose 3} + {n \choose 5} + \, \, ...$$ using the "weirdo" method (i.e., where one of the elements is chosen as special and included-excluded -- I'm sure you get the idea).  After days of repeated effort, the proof has failed to strike me. Because every time one of the elements is excluded, the term would be ${n-1 \choose k}$ and not $ {n \choose k}$, which is not the case in either of the sides of the equation. Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$ Prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does not Exist using Polar Coordinates Let $h(x,y)=\frac{x^5y}{2x^{10}+y^2}$. How would I prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does Not Exist?  I think that we might use polar coordinates, but I am not sure. I used to use a lot ctrl+alt+numpad to move windows, how can I use this feature on ubuntu 18 like it was default before?  Thank you in advance How do you debug PHP scripts? How do you debug PHP scripts?  I am aware of basic debugging such as using the Error Reporting. The breakpoint debugging in PHPEclipse is also quite useful.  What is the best (in terms of fast and easy) way to debug in phpStorm or any other IDE? What is the best choice for .NET inter-process communication? Euler-Lagrange equation in curved spacetime: how to calculate $\partial \mathcal{L}/\partial(\nabla_\nu A_\mu)$? My problem is mostly with notation. When we apply the Euler-Lagrange equation to the Lagrangian scalar $$\mathcal{L}= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$ Where $$F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu= \nabla_\mu A_\nu -\nabla_\nu A_\mu$$ and I start with $$\frac{\partial\mathcal{L}}{\partial A_\mu}=\nabla_\nu\left(\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}\right).$$   Now I rewrite the lagrangian as $\mathcal{L} = -\frac{1}{4}g^{\mu\alpha}g^{ \nu\beta}(\nabla_\mu A_\nu-\nabla_\nu A_\mu)(\nabla_\alpha A_\beta-\nabla_\beta A_\alpha)$ $$\mathcal{L}= -\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}\left[(\nabla_\mu A_\nu)(\nabla_\alpha A_\beta)-(\nabla_\mu A_\nu)(\nabla_\beta A_\alpha)-(\nabla_\nu A_\mu)(\nabla_\alpha A_\beta)+(\nabla_\nu A_\mu)(\nabla_\beta A_\alpha) \right]$$ My problem is calculating $\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}$, how are the rules of partial derivatives generalised to covariant ones? Also, is $\frac{\partial\mathcal{L}}{\partial A_\mu}=0$, is that correct ? How to combine paths in Java? Is there a Java equivalent for  in C#/.NET? Or any code to accomplish this?  This static method combines one or more strings into a path. What is "angle of view" in photography? I read on the :     Focal-length determines the angle-of-view seen through a lens for a   given sensor-size. With a full-frame sensor a lens gives the same   angle-of-view as it would on a 35mm film camera. With a smaller   sensor, the angle-of-view becomes smaller. The crop-factor, also   called FLM, is the ratio representing the difference in equivalent   focal-lengths. So a 150mm on a full-frame DSLR such as the Nikon D700   gives the same angle-of-view as a 100mm on a D7000 since its FLM is   1.5X.      Short focal-lengths show a greater angle-of-view compared to longer   ones. [...]   In layman's terms, what is angle-of-view? Is it the same as focal length? If not, how is it different? How is it used? Why do I need to know about it? How do you clean a seasoned cast iron skillet? I have a seasoned cast iron skillet, and I don't want to mess up the seasoning when I'm cleaning it.  What do I use and what don't I use to get it back to clean?  I've heard not to use soap and to make sure it is dry, but nothing beyond that. While using MathJax to typeset binomial coefficients, I came across this problem of different sized brackets if my lower index contains the '0' character. Is there anyway to make the brackets the same size? Do I cast the result of malloc? In , someone suggested in a  that I should not cast the result of malloc. i.e., I should do this: int *sieve = malloc(sizeof(int) * length);  rather than: int *sieve = (int *) malloc(sizeof(int) * length);  Why would this be the case? How to find a good topic for a PhD research proposal? In many countries an application for a PhD position includes a written research proposal, so my questions is what are some advises/strategies to come up with a good topic/idea for a PhD research proposal and how can one assess the quality/fruitfulness of an idea? As an undergraduate student one just doesn't have the experience to foresee which ideas might have promising research results and which probably won't have. (And I doubt that potential supervisors have the time to comment on every idea of every potential applicant in cases where it is possible to establish some kind of contact before the actual application.) var functionName = function() {} vs function functionName() {}