Is it possible to find two disjoint subsets $X$ and $Y$ of $\mathbb{R}$ such that both are dense in $\mathbb{R}$ and both are locally uncountable?  By a locally uncountable set $X \subset \mathbb{R}$, I mean a set which has the property that if I take any nonempty open subset $U$ of $\mathbb{R}$, then $U \cap X$ has cardinality strictly larger than the natural numbers.
First, this question is motivated by the imprecise question: Is there a sensible notion of parity (evenness and oddness) for real numbers?   Here are some properties a notion of parity for $\mathbb{R}$ should have:    It should be an equivalence relation on $\mathbb{R}$ with exactly two equivalence classes.  Each equivalence class should be dense in $\mathbb{R}$. There should be some kind of symmetry between the two equivalence classes. (This is intentionally imprecise.)   If we divide $\mathbb{R}$ into rationals and irrationals, this lacks symmetry, since the rationals have measure zero and are countable, while the rationals have infinite measure and are uncountable.   So we could require that each equivalence class be uncountable and/or have positive (or infinite) measure. Then take one equivalence class to be the rationals, union with countably many fat cantor sets with positive measure, and the other class its complement. Then both classes are now dense, uncountable, and have infinite measure. But again, symmetry is lacking. One of these classes is "uncountable everywhere" and has "positive measure everywhere" while the other does not.   I propose some definitions: A subset $A \subset \mathbb{R}$ is uncountable everywhere if for any open interval $(a,b)$, the intersection $A \cap (a,b)$ is uncountable. A subset $A \subset \mathbb{R}$ has positive measure everywhere if for every open interval $(a,b)$, the intersection $A \cap (a,b)$ has positive measure. We can see immediately that having positive measure everywhere implies being uncountable everywhere, since a positive measure set must be uncountable.   Finally, my questions:    Is there a partition of $\mathbb{R}$ into two sets that are both uncountable everywhere?  Is there a partition of $\mathbb{R}$ into two sets that both have positive measure everywhere?  Is there a partition of $\mathbb{R}$ into two sets that split each interval $(a,b)$ into two parts of equal measure?   Potential generalizations for extra credit: What about partitions of $\mathbb{R}$ with $n$ equivalence classes, where $n \in \mathbb{N}$, or even with countably many, or even uncountably many equivalence classes?  Note: If you've seen these definitions of uncountable everywhere or positive measure everywhere somewhere under a different name, please let me know. I've never found anything where other people were thinking about these notions.  EDIT: By "measure," I mean Lebesgue outer measure, so we don't have to worry about anything being measurable.
I am trying to prove $(a)$ is a maximal ideal in $\mathbb{Z}$,  if and only if $a$ is prime number.  Now I wrote: assume $a \in\mathbb{Z}$, while it's not prime number we can write as $a=xy$, for some integers $x$ and $y$. then $(a)\subset (x)$ and $(a)\subset (y)$  while if $(a)$ is maximal ideal, it's not exist $(k)$ such that $(a)\subset (k)\subset\mathbb{Z}$. so $(a)$ cannot be maximal ideal. we can say by contradictory if $a$ is prime, $(a)$ is maximal ideal.  is this correct?  and how to prove the other way. At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) \neq -g(x)$. It follows that $f$ is homotopic to $g$.  My idea is to use the mapping cylinders $M_f, M_g$ to construct a homotopy via $$M := M_f \amalg M_g/ (x, 0)_f \sim (x,0)_g$$ But this doesn't seem to work. In addition I don't know how $f(x) \neq -g(x)$ is going to help me (I can see how this could help if $n=1$, but I have no clue how this would work for $n&gt;1$).  I would appreciate any hints towards a solution.  Thanks in advance. In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now. Redirect all requests to HTTPS, except for one subdirectory How would one be able to prove mathematically that $1+1 = 2$? Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically? gpgkeys: key C927EBE00F1B0520 not found on keyserver Possible Duplicate:    I am using 10.04 LTS - Lucid Lynx. I shutdown the computer while a software update was running. Then after I restarted my computer and while the update manager was running, I got this error:  Could not download all repository indexes The repository may no longer be available or could not be contacted because of network problems. If available an older version of the failed index will be used. Otherwise the repository will be ignored. Check your network connection and ensure the repository address in the preferences is correct. GPG error: http://toolbelt.herokuapp.com ./ Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY C927EBE00F1B0520 Failed to fetch  http://ppa.launchpad.net/rabbitvcs/ppa/ubuntu/dists/**DISTRIBUTION**/main/binary-i386/Packages.gz 404  Not Found   I went through the answers to  and I tried this $ sudo launchpad-getkeys  Please wait... launchpad-getkeys is running an update so it can detect the missing GPG keys Trying to import all the missing keys gpg: requesting key 0F1B0520 from hkp server keyserver.ubuntu.com gpgkeys: key C927EBE00F1B0520 not found on keyserver gpg: no valid OpenPGP data found. gpg: Total number processed: 0 launchpad-getkeys has finished importing all missing GPG keys.  Try running sudo apt-get update - you shouldn't see any key  errors anymore  But when I run sudo apt-get update again, it again shows that the key was not found on the keyserver. How to solve this problem? Giving edit reputation bonus for same user for same post multiple times On Stack Overflow, I saw some users editing the same question multiple times, and getting the 2 point reputation bonus for each time.  I saw it during review of a suggested edit (some months ago, I forgot the question that's why I'm unable to paste the link here). I rejected it and gave the reason in the "Other" reason field.  Most of the time I saw that users first formats the code, and then after approval they edit the title and fix the grammar.  Why is this allowed?  My point is, if an user gained 2 reputation points for a particular edit, then why give another 2 points again for the same post? Because the second edit on same question/answer means that there were some mistakes or misses in their previous edit. My equation is not showing big bracket and not showing in next line Where do students who don't get into Hogwarts go? Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field Conditionally use 32/64 bit reference when building in Visual Studio Why is my membership length only "Six months"? I was viewing the  and I noticed that I've apparently only had my account for six months, which is news to me... I opened my first site account here in February of 2014.  My account is much older than this, what's causing this bug?  To add some detail... On the per-site list, it does line up the &quot;member for X&quot; text with my actual membership time on that site. For example, when I look at the Meta Stack Exchange leaderboard, it shows me as a member for 2 years, 10 months. This is expected.  Update. With some work, today, I've managed to exceed my Interpersonal Skills hat count of with my Meta Stack Exchange hat count. This has changed my account age to match my MSE account age...  Still not reflective of my actual age... but better. Send mail without MFMailComposeViewController I am using \tableofcontents to create TOC in my document. however it is appearing as shown in the image. As the numbers increase the gap between the number and text is reducing.   Thanks for your help Aku Can wizards really cast spells in armor? Check if table exists in SQL Server I would like this to be the ultimate discussion on how to check if a table exists in SQL Server 2000/2005 using SQL Statements.  When you Google for the answer, you get so many different answers. Is there an official/backward and forward compatible way of doing it?  Here are two possible ways of doing it. Which one among the two is the standard/best way of doing it?  First way:  IF EXISTS (SELECT 1             FROM INFORMATION_SCHEMA.TABLES             WHERE TABLE_TYPE='BASE TABLE'             AND TABLE_NAME='mytablename')     SELECT 1 AS res ELSE SELECT 0 AS res;   Second way:  IF OBJECT_ID (N'mytablename', N'U') IS NOT NULL     SELECT 1 AS res ELSE SELECT 0 AS res;   MySQL provides the simple   SHOW TABLES LIKE '%tablename%';    statement. I am looking for something similar. Error when updating texlive 2015 When I want to update packages of texlive 2015 I get this error? what is the problem?  tlmgr update --list  The locale codeset (cp720) isn't one that perl can decode, stopped at   C:\texlive \2015\tlpkg\tlperl\lib/Encode/Locale.pm line 123. Compilation  failed in require at C:\texlive\2015\tlpkg\tlperl\lib/LWP/UserAgent. pm  line 1001. Footnote without a marker Beamer: How to Disable `\pause` Temporarily Concatenate all arguments and wrap them with double quotes