Question. Is there a Hausdorff space $ X $ such that every Hausdorff space $ Y $ can be embedded into $ X^\Lambda $ (given the product topology) for some sets $ \Lambda $?  Concerning Tychonoff spaces, $ X = [0, 1] $ satisfies the condition; i.e., every Tychonoff space can be embedded into some powers of $ [0, 1] $. I wonder if there is a counterpart for Hausdorff spaces.  I tried to disprove the existence by proving that for every Hausdorff space $ X $ there exists a non-trivial Hausdorff space $ Y $ such that there are only a few continuous maps from $ Y $ into $ X $, but I got stuck.
There is a nice property of normal spaces, namely, closed disjoint subsets can be separated by continuous functions into $\mathbb R$.   Then you ask yourself, what about Hausdorff spaces?, are all Hausdorff spaces completely Hausdorff?   Well, if we choose our separating space $X$, that is, the space such that for any Hausdorff $Y$ and any distinct $a,b\in Y$ there is some continuous $f:Y\rightarrow X$ with $f(a)\neq f(b)$, to be completely regular and Hausdorff, this cannot happen by an example of E. Hewitt, Hewitt’s Condensed Corkscrew, call it $\mathbb H$; as any continuous function from $\mathbb H$ into $\mathbb R$ is constant, and any Hausdorff completely regular space $X$ can be embedded into a product of closed intervals $\prod_{i\in I}[0,1]$, and thus any continuous $f:\mathbb H\rightarrow X\subseteq \prod_{i\in I}[0,1]$ is constant, as each $\pi_i\circ f$ is constant, so our candidate to be the separating space $X$ cannot even be $[0,1]$.  My question is, does there exist a Hausdorff space $X$ such that for all Hausdorff spaces $Y$, and any distinct $a,b\in Y$ there exists a continuous $f:Y\to X$ such that $f(a)\neq f(b)$?  Thanks.
I've seen a lot of games in the LCS and All-Star, and they sometimes changed bot and top lanes.   When is this a good practice? What do you gain from doing this? Should I do this to avoid counters or if my solo laner has more sustain than theirs? How to blur a texture node in cycles material? For a Cycles material I need to blur a texture. If the texture where a Texture Image node, I can blur the image before adding it but in my case I need to blur a Voronoi Texture and so I can't do it.  I have searched but I didn't find a blur node. Can this be achieved? I had this problem a week or two ago and I posted here but nothing from the suggested solutions seemed to work. I have managed to get some more details on it, in case anyone can help me, because this issue is causing me huge productivity problems and I really want it solved.  So I'm using Linux mint distribution, and I've installed Lyx via the software manager. Every time that I open a new document, I get an error claiming that some files are missing, and until they are installed, compiling the files or exporting them as pdfs is not possible. The file in question is called "extarticle.cls". Additionally, under Document->Settings->Document Class, every document class is marked as unavailable.  Here is the thing though.. this file is not missing... I have located it inside a folder associated with Lyx. It's definitely not missing.  I really need help with this one guys... It's been weeks since the problem came up and I really just want to be rid of it.. Why does this thing not work? Any help? Why is my profile image different? When I first signed up on a Stack Exchange site and got a profile image, it was this one (see my current user card I'm actually not sure what I look like any more):     Later, I got  and SEDE profiles, and my user image there was a different identicon. Because . I would have a picture of it, but that's the problem. ... Thanks to , I have a picture:     On those two sites, it changed from that purple one to this new one recently:   I don't know when the change happened exactly, but I KNOW I didn't change it (I don't know how to change it, otherwise I would have changed it to my main pink one).  Why did this change? Can I get my other icon back or will I be stuck with this new one?   Also, are the gravatars at Area 51 and SEDE securely generated?    After looking in my profile settings, I really confused (I am seeing my wrong avatar in more and more places since I first posted this.):        Seriously.......    What...?    (I'm using this , but be warned that they will try to IP block you after a certain number of requests to force you to pay... May look for a better alternative later.) Why would spacetime curvature cause gravity? It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force?  Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this?  When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place? Is it possible to trade Pokemon from the 1st generation games to the most recent ones? uint color;  bool parsedhex = uint.TryParse(TextBox1.Text, out color);  //where Text is of the form 0xFF0000 if(parsedhex)    //...   doesn't work. What am i doing wrong? Cartoon TV Show About Three Kids and Monsters and a "veil" between realms Is it possible to upgrade a BGA 1224 CPU? The impression that I got from the  films was that if  was successful, most of the world was going to suffer greatly. Basically everyone would serve Voldemort as slaves or die. So what's the point of serving someone who's probably going to ultimately end up betraying and/or enslaving you? I know the  hated , but is that really enough to justify tearing down Wizard society? Why exactly do the Death Eaters serve Lord Voldemort? Fear? Is the post-Voldemort world not that bad? (I for one welcome our ) How to monitor who is pinging me? Is there a way to know who is pinging my PC on my network? Is it possible to monitor all ICMP packets? How can I do that? Can you create an illusion of empty space? I've been thinking of various ways for a crafty Illusionist/Rogue to 'hide in plain sight' by essentially creating the cover that they want to hide behind.  Ideas include things like:   Create a wall that is 2.5ft closer than it really is and hide in said space Create some innocuous barrier or container and hide within it Create a 'sheet' that essentially has what an empty room looks like 'painted' on it, so someone walking past an open door would see an empty room.   From different discussions I've had, these all should work, but the one lingering one I have is, what if my illusion is simply 'empty space'?  If I were to create an illusion of empty space and then place myself in it, would I be essentially invisible?  What if I created an illusion that was partially-empty space, like a 5x5 cube of swiss cheese? Would it seem like there was a swiss cheese cube formed around my person or could my 'fake-empty-space-image' fill the holes?  Would my 'empty space' illusion simply need to work like the aforementioned 'sheet of what I expect you to see' in order to function (and thus most likely only work from limited viewing angles)? I want to create a custom control in C#. But every time I have to fully redraw my control, it flickers, even if I use double buffering (drawing to an Image first, and blitting that).  How do I eliminate flicker when I have to fully redraw? Started new job. Rollover previous employer 401k to new 401k, IRA or Roth IRA? I did apt-get upgrade it never finishes at this point:  Searching for GRUB installation directory ... found: /boot/grub Searching for default file ... found: /boot/grub/default Testing for an existing GRUB menu.lst file ... found: /boot/grub/menu.lst Searching for splash image ... none found, skipping ... Found kernel: /boot/vmlinuz-3.2.0-24-virtual Updating /boot/grub/menu.lst ... done  You already have a NoLOADER configuration in /etc/grub.conf Running boot loader as requested Running /usr/sbin/grub  ...    The system does not freeze (Ctrl-Z works, I get my console back) just that that /usr/sbin/grub seems to never complete.  What should I do? Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits I have just proved the divisibility rule for 3 and 9.  Let $n\in\mathbb{N}$. Let $m$ be a number that appears when you shuffle the digits in $n$. Show that $n-m$ is a multiple of 9.  Can anyone offer some help? Straighten Vertices Along a Line On a line, I need to line up the bottom vertex points in relation to a top reference vertex point. How could I reset the X Y and Z coordinates of these vertex points to essentially line them up all through use of the vector of the reference point vertex.  I do not have access to Mesh -> Transform -> Scale on Axis for aligning vertices along a line. How can I permanently enable cheating? Integral of sinc function multiplied by Gaussian I am wondering whether the following integral $$\int_{-\infty}^{\infty}  \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If analytical form does not exist, I really appreciate any alternative means for evaluating the integral. One method would be numerical quadrature including Gaussian quadrature, but it may be inefficient when the parameters $a$ and $b$ are very different in scale.  EDIT : In view of  discussion, we have decided to add OP's self-answer to the end of the question, for it does not qualify as an answer yet contains vital details. The copying is unabridged. Thanks very much for your comments, and the following result was obtained including the case for $x_0 \ne 0$: $$ \int_{-\infty}^{\infty} dx \exp[-a(x-x_0)^2] \frac{ \sin(bx) }{ x } = \pi \exp(-a x_0^2) \mathrm{Re}\left(\mathrm{erf}\left[\frac{b+2iax_0}{2\sqrt{a}}\right] - \mathrm{erf}\left[\frac{2iax_0}{2\sqrt{a}}\right]\right) $$ where $a\gt0, b, x_0$ are assumed to be all real. (note: coefficients etc may be still wrong...) This integral appears in a type of electronic structure calculation based on a grid representation (sinc-function basis). I believe the above result should be definitely useful. Thanks much!! --jaian When is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation? When reading to an audience, or speaking in conversation, when is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation (assuming you know the appropriate pronunciation for it)? Is it considered rude, or condescending? Or is it considered a mark of being knowledgeable?  One of the things to consider are place names. When referring to Paris, France, should it be pronounced with the silent 'S' as the French would say it, or with the 'S'? Should Hiroshima be pronounced as a Japanese speaker would pronounce it slightly more emphasis on RO, rather than the SHI? Should Mount Pinatubo be pronounced as a Filipino speaker would pronounce it with shorter stronger vowels, or the longer vowels? (e.g. Pi as is 'pick' rather than 'pea'.) Should Wichita be pronounces as the original "shi" rather than the modern "chi"?  Pronounce pesos or sombrero as a Spanish speaker 'eh' or the English 'ay'?  (I can't think of other common words right now that aren't words taken from other languages like hurricane, boondocks, tornado, etc. which I think have (correctly) changed to English pronunciations.)