I already know that if $X$ is a compact metric space then the space of continuous real valued functions $C(X \to \mathbb R)$ are separable. What I'm trying to prove that if $X$ is a locally compact metric space and is $\sigma$-compact. The space of continuous compactly supported functions $C_c( X \to \mathbb R)$ are separable.    Here's my approach: As $X$ is $\sigma$-compact, we can write $X = \bigcup_{ n=1}^\infty K_n$, where $K_n$ are compact and increasing. Then since each $C(K_n \to \mathbb R)$ is separable, it suffices to show that the set $ \{ f \in C_c (X \to \mathbb R) : \operatorname{supp}(f) \subset K_n \}$ is dense in $C(X \to \mathbb R)$ (Not sure). Then get I stuck.  Any help is appreciated.
Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable?  My work so far:   If $X$ is a compact metric space, then by Urysohn's Lemma and Stone-Weierstrass, the continuous functions $C(X)$ on $X$ are separable and hence the result follows as $C_{c}(X) = C(X)$.  Suppose $X = \mathbb{R}$. Write $\mathbb{R} = \bigcup_{N  = 1}^{\infty}[-N, N]$. Let $f \in C_{c}(\mathbb{R})$. Then $f$ is supported on a compact set $K \subset [-N, N]$ for some $N$. Thus $C_{c}(\mathbb{R}) =\bigcup_{N = 1}^{\infty}C([-N, N])$. Each $C([-N, N])$ has a countable dense subset $\{\psi_{N, n}\}_{n = 1}^{\infty}$ and so $\bigcup_{N, n = 1}^{\infty}\{\psi_{N, n}\}$ is a countable dense subset of $C_{c}(\mathbb{R})$.  In the general case, $X = \bigcup_{i = 1}^{\infty}X_{i}$ where each $X_{i}$ is compact and $X_{1} \subset X_{2} \subset \cdots$. Let $f \in C_{c}(X)$. Then $f$ is supported on a compact set $K = \bigcup_{i = 1}^{\infty}K \cap X_{i}$. Is it still true that $C_{c}(X) = \bigcup_{i = 1}^{\infty}C(X_{i})$?
Ubuntu 20.04 still supporting Python2 How do I check if someone accessed my computer? I need to check the list of files accessed in my computer (ex: yesterday on particular time). Is it possible?   Or   I need to check whether anybody accessed my computer (when I left it unlocked) or not for a particular time. The equivalent electric field of a magnetic field I know that Lorentz force for a charge $q$, with velocity $\vec{v}$ in magnetic field $\vec{B}$ is given by  $$\vec{F} =q \vec{v} \times \vec{B}$$  but there will exist a frame of reference where observer move at same velocity with that of charge $q$, so according to him $v=0$. hence he will see no magnetic force is exerted on charge $q$. I have work on this problem for a while and found that the special relativity predicts equivalent electric force will acting upon charge instead. I want to know the relationship between this equivalent electric force and magnetic force.  Thanks in advance How prove this equation $\alpha=\aleph_{0}\alpha$ Question:     let $\alpha$ be a infinite cardinal , show that:   $$\alpha=\aleph_{0}\alpha$$   where $\aleph_{0}$ is the cardinality of the natural numbers is denoted aleph-null   this equation is from a book,and I can't prove it,and the author say this is important equation,and can't prove it.Thank you for you help Remove a child with a specific attribute, in SimpleXML for PHP I have several identical elements with different attributes that I'm accessing with SimpleXML:  &lt;data&gt;     &lt;seg id="A1"/&gt;     &lt;seg id="A5"/&gt;     &lt;seg id="A12"/&gt;     &lt;seg id="A29"/&gt;     &lt;seg id="A30"/&gt; &lt;/data&gt;   I need to remove a specific seg element, with an id of "A12", how can I do this?  I've tried looping through the seg elements and unsetting the specific one, but this doesn't work, the elements remain.  foreach($doc-&gt;seg as $seg) {     if($seg['id'] == 'A12')     {         unset($seg);     } } I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier? Matrix representation of linear transformation How can I migrate a Windows 10 from BIOS/MBR boot to UEFI/GPT without reinstalling? Incorrect icons on hot network questions I see the same problem has occurred in the past, the most-linked one being   Is this happening for anyone else?   I've marked the icons I could recognize as incorrect.  This exists on every page, is on both chrome and firefox (from Ubuntu 16.04), and persists after hard-refreshes in each.  Edit: The icons in Meta's Hot Network Questions tab appear to be consistently accurate. Every other page I checked (including Unix, Workplace, English Language &amp; Usage, Money, and Sci-fi) have a handful of incorrect ones, even for some of the same questions that are displayed correctly in Meta.  Further note: if it matters, the icons are also incorrect in the drop down community-search feature on the right end of the header. Again, they appear correct on this Meta page, and incorrect on all other pages I've checked. Some of the ones that are consistently incorrect:   computational science constructed languages movies and TV raspberri pi rpg russian language science fiction and fantasy sitecore skeptics super user latex workplace   That distribution seems skewed, but I may just be disproportionately familiar with the sites at the end of the alphabet. Many others could be wrong, those are just the ones I could recognize.    Last edit: apparently my Google skills were lacking -  was asked an hour prior. Solve differential equation, $x'=x^2-2t^{-2}$ Solve differential equation:  $x'=x^2- \frac{2}{t^2}$  Maybe is it sth connected with homogeneous equation? I have no idea how to solve it. ReadList can't read from file containing high-precision Reals Does a Nikon DX lens on a FX camera get subjected to the 1.5X crop factor? I know that there is a very similar question, but I'm not interested in highlighting. What I'm looking for is some tool in Linux to edit pdf files. I've tried pdfedit which did the work for me, but if freezes after a few edits. I've tried this tool under Ubuntu 9.10 and under KDE Fedora 14 and the same thing happens. I've tried Okular but the notations are too big to fill out exercises from pdf workbooks.    Is there any tool that suits this job? I don't care whether it is open source or I have to pay a reasonable price for it. Is there a general formula for image size vs. print size? I tried to choose a answer for a question on SA and someone immediately answered with the correct response. I try to click the check box and it says wait 2 min. I keep clicking until it changes from 2 min to 1 min. I wait 60 seconds on my watch and click again and it says please wait 59 seconds.  The 2 minute mark should have said 3 and the 1 minute mark was really 2.  It seems that this has , which is a lot more inconvenient than 59 seconds. Logical Volume, Physical Volume and Volume Groups How to receive sms from our android app? I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? Add "Delete with reason" button/dialog to existing moderator tools for flagged answers While this is an old problem, it's one that will only increase with time.  As Stack Exchange sites gain in popularity, more and more users are coming to the sites in search of answers to their specific problems.  Unfortunately, not all of them understand the basic operating principals behind Stack Exchange, and ask questions, or start discussions as separate answers.  Deleting these answers is fine, but more often (I would argue almost always) than not, a new user doesn't understand why.  Granted, comments are a way of doing this, but there are drawbacks to comments:   They require extra, replicated effort on the part of the moderator for something that they do repetitively The message will be inconsistent; messages like this would be consistent, cite the relevant resources that guide the user in the right direction The larger size would have a higher chance of grabbing the user's attention, as well as put more weight behind the action (as it's officially attached to the answer in a consistent manner). The banner will serve as an example of what not to do to other users who can see the deleted post.   Additionally, there are tools that can help with auto-commenting on posts, but these have their drawbacks as well:   Not all moderators use these tools The messages that are delivered by the tools are ultimately generated initially by the moderators, so the message will not be consistent These tools are not always available on other devices (e.g., mobile platforms)   It would be helpful to the user if they had a reason indicating why the answer was deleted.  Something along the lines of:    In order to accomplish this, I propose an additional (not a replacement) button in the moderation tools labeled "delete with reason".  Mods would still be able to delete answers outright, they just have the option now of stating why in an easy manner which will allows for more time to moderate harder.  Placement is not important (screen real estate is valuable, yes), but I envision it looking something like this in the moderator tools:    When clicking "delete with reason", a dialog, much like the "close" dialog will appear with options:    Initial options for the reasons could be:   Asking a new question Starting discussion   A mechanism like this has the potential for other benefits as well:   Users with an answer deleted with a reason might actually be directed to perform the right action now, and in the future (creating better site denizens) Users who wish to contest the deletion of an answer on meta can cite that specific reason, which mods can specifically address (which actually might reduce noise on meta about "why was my answer deleted?") Can reduce flags on answers asking "why was this deleted?" (admitted, I've not personally experienced this to be a pervading problem on Stack Overflow, but this might be the case for other mods on other Stack Exchange sites) Other Stack Exchange sites which don't have that many users will give a better experience to their users when their first action isn't ideal for a Stack Exchange site; this will enhance the chance that the site will retain the user and get good contributions from them (as per Farseeker's comments: , ).   Additionally, if this works well for the moderator tools, I imagine that it would not be such a leap to eventually have this graduated to users with the rep to cast delete votes, thus allowing non-mods to deliver a more clear message to new users about how to best participate on a Stack Exchange site. Method always returns true?