Down scaling images with Pygame
How can you draw more detailed/smoother images in pygame?
If a topology contains all infinite subsets, then it is the discrete topology I am solving question $8$ of exercises in section 1.1. of chapter 1 in "Topology without tears". The question reads as follows.     Let $X$ be an infinite set and $\tau$ be a topology on $X$. If every infinite subset of $X$ is in $\tau$, prove that $\tau$ is the discrete topology   My idea was to show that every singleton set is in the topology and thereby the topology is discrete.  Choose an infinite set $A$ from $X$ such that $A^c$ is also infinite. $A$ and $A^c$ belong to $\tau$ since they are infinite sets. Now consider any $x \in X$. Note that $A \cup \{x\}$ and $A^c \cup \{x\}$ belong to $\tau$ since they are infinite sets. Hence, their intersection which is nothing but $\{x\} \in \tau$ for every $x \in X$.  The problem I have is I don't know how to prove the first sentence in the previous paragraph. These are my line of thoughts to prove them.   If $X$ is a countably infinite set, then I can list the elements are let the odd numbered elements fall into $A$. This guarantees $A$ and $A^c$ are both infinite. If $X$ is uncountable, then I can choose a countably infinite subset and call it $A$.   Are the above arguments rigorous? I am not especially happy with my second argument of choosing a countably infinite subset from an uncountable set since I do not give an explicit procedure of constructing the set $A$.  Also, is there any other simpler way of answering the original question? Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$  According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\ell^2$-norm, but there is no proof. How can I show that $C$ is indeed compact in $\ell^2$ ? How to delete kexts in Catalina? I have a 2014 MacBook Pro experiencing . Catalina does not allow kexts to be deleted via this method. Unless I am able to delete or disable this kext my laptop will shut down randomly. Anyone know how to remove kexts in Catalina? Built-in unit strings recognized by Quantity? A lot of people will probably gloss over this fact, but it looks like C-3PO meets BB-8 for the first time in this scene 0:20:                                But how does C-3PO know BB-8's name? Is there a time where C-3PO meets BB-8 before this and knows his name? Do lenses have focal planes or focal spheres/ellipsoids? How can I use variables in the LHS and RHS of a sed substitution? How is memory allocated for an implicitly defined multidimensional array in C99? Print escaped representation of a str How do I disable the guest session in Ubuntu 11.10 or higher? I don't want people to be able to use my computer without using a password to log in! Displaying page construction guides Is there any way to overlay some lines and dimensions on the document so that one can see where the margins end and more easily examine the overall layout and ratios of elements on the page?  See for example, some of the illustrations of frameworks in this article on the  and this one on  (which displays dimensions). How do we "restore" notepad after crash? I was running NotePad and Chrome when my computer suddenly shuts down.  After reboot, chrome was able to "restore" itself such that I didn't lose any data.  However, how do we "restore" notepad.exe? 'Should've seen it glow' or 'should've seen it glowing'? Which one of the following is the correct one?   I should have seen it glow. I should have seen it glowing.   Or are both correct? Would you parse them please? When would you use the Builder Pattern? What are some common, real world examples of using the Builder Pattern?  What does it buy you?  Why not just use a Factory Pattern? Why isn't $V= V_1 + V_2$? $V=V_a - V_c = V_a - V_b + V_b - V_c$, $V_a - V_b= V_1$ and $V_b - V_c = V_2$ Doesn't that prove that $V = V_1 + V_2$? Regardless of $V_3$, If i'm wrong , is there a way to obtain $V$ in terms of $V_1$, $V_2$ and $V_3$ I don't know which buttons I have to click or what key combination to type to calculate a cube root. Reset post score to 0 on migration I suggest that when a post is migrated from one site to another, that the question vote score get reset to zero (or perhaps only a negative vote score is reset to zero). Often people will downvote a question when the only thing that's "wrong" with it is that it's posted on the incorrect site. After migration, those downvotes hang around as a negative score and may unfairly bias the treatment of the question. There is a  being thrown out of a flask, literally hovering out of the container  My question is how detailed a physical explanation of this phenomenon we have?  Question: More precisely, if we used the best available mechanical physics simulator software, and we tried to simulate this behavior, would the software be able to reproduce the effect or not? My guess is not but I would like to know why not.  Or do we have to content ourselves only with handwaved explanations? I am from the school of thought that you don't understand physics until you have a reasonable certainty that you can reproduce the physics in a computer. What are the advantages of a custom ROM over rooted stock? Can a set be neither open nor closed? An example would do. I cant think of any.   Thanks in advance!