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I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this.  Let $X$ be a set of uncountable cardinality. Using the Principle of Well Order we have a well ordering $\le$ on X(and $&lt;$ means $\le$ but not euqual). By adding an element, which we denote by $∞$, and introducing the convention that $x &lt; ∞$ for all $x ∈ X$, we will assume that X has a maximum with respect to $\le$.   We define $ω_1 = \text{min} \{ x ∈ X : \{y ∈ X : y &lt; x \} \  \text{is uncountable}\} $  Clearly such a $ω_1$ exists. we define $0 = \text{min} \ X$ and intervels $[0,x], [x,y), (a,b]$ etc. in the usual sense.  Here are my questions:  (1)For any countable $A ⊂ [0, ω_1)$ there is an $x &lt; ω_1$ so that $A ⊂ [0, x]$.(I don't know how to make use of the countability and uncountablity here)   (2)Equipped with the topology generated by open intervals,  $[0, ω_1]$ is compact.   (3)A famous application of the first uncountable ordinal is to find an example of a Borel measure (with respect to the topology in (2)) that is not Radon. So how to construct a finite Borel measure $µ$ on $[0, ω_1]$ which is not a Radon measure?  Any solutions or elementary references will be appreciated! how to have a custom zsh auto_cd command Instead of the normal cd I've created my own function  function cd {     builtin cd "$@" &amp;&amp; ls -F }   This works great for when I use commands like cd someDir/subDir but not when I change directories via zsh's auto_cd someDir/subDir.  Is there a way to customize what command gets called for auto_cd? "...his parents' dream of *him* achieving a Cambridge degree." What is the function of "him" here? Why is notepad++ writing to my new document? I updated notepad++ and it started writing to the screen.  I quickly closed it not knowing what it was, but now I'm curious do I have a 'hacked' copy of notepad++? I play D&amp;D 5e and sometimes come here with questions. But sometimes I see things referenced here as sources that I don't have or aren't aware of....  Which are the "official rules" for D&amp;D 5e, and where do I find them? New breaker and panel: What's the brown goo where they connect? I find I have to scroll a lot to move the text of my page compared to Windows where I can choose the speed of the mouse wheel.   Does Ubuntu have a similar utility and if so how can I access it? When is the image of a linear operator (between Banach spaces) closed? Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces.  When is $\operatorname{Ran}(T)$ a closed subspace?  What theorems are there?   Thanks :) What exactly is an environment variable? Split Data Based on Year in a Visual force Page PDF Safari Extension to Manage Tabs? I am having difficulty with this proof. I have previously proved that for an automorphism on the extension field $K$ of $F$ that when restricted to $F$ is the identity automorphism must take roots of $f(x)\in F[x]$ in $E$ to roots of $f(x)$ in $E$. I was thinking of using the minimal polynomial $x^{3}-2\in\mathbb{Q}[x]$ and the result of previous problem I solved, but I suspect there are automorphisms that don't fix the subfield, so the result can't be used or I need to prove that any automorphism of an extension field must fix the subfield that is being extended or always sends roots in $E$ to roots in $E$.  I am not asking for a solution to this, but maybe a hint or pointer in the right direction. This problem is introduced in the text prior to Galois theory, so please keep that in consideration. I'm sorry if this is a broad question but can someone explain to me how to interpret these regression diagnostic plots? I understand the Normal Q-Q show's how normal the spread of the data is, but the other three I am confused by. For the Residuals vs Leverage, what does it mean for a point to be outside of the bounds? What does it mean if the red lines are parabolic and curved like in the first two? Why do we care about residuals, and what do they tell us? I've searched the web multiple times for some guide to interpret these and I can't get any straight forward answers. Any help is appreciated, thank you. Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ by defining a bijection $f:A^{B \cup C} \rightarrow A^B \times A^C$.  Any hints on this one?  Thank you! If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true. My problem: Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.  I tried like this:  Suppose S is LI.Then the vector $0$ cannot be expressed as a linear sum of all elements of $S$.  How it follows that a finite subset is also LI from this fact. I think $S$ can be finite or infinite.  This is a question from the book Linear Algebra - Friedberg et al. Prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$.   Do I use the terms $x= r^2 - s^2$, $y = 2rs$, and $z = r^2 + s^2$ to prove this problem?  Thanks for any help. I have an unused copy of Mass Effect 2 Collector's Edition bought via Amazon.com in the US, including a special code for Inferno Armour (also unused).  Will it work fine if I install and start playing the game in the UK? Does geographical location matter?  Thanks! What is the VC dimension of a decision tree? What is the  of a decision tree with k splits in two dimensions? Let us say the model is CART and the only allowed splits are parallel to the axes.  So for one split we can  and then for any labeling of the points we could get perfect prediction (i.e.: shattered points)  But what about 2 splits, or any general k? martingale and stochastic Integral