I've seen a proposition like that Proposition: If $f:X\longrightarrow Y$ is an open and continuous function and $D$ is dense in $Y$, then $f^{-1}(D)$ is dense in $X$. Is that proposition correct? if so, here is my approach : if $D$ is dense in $Y$, then for all nonempty open $G$ in $X$, $f(G)\cap D\neq\emptyset$ then $f^{-1}(f(G)\cap D)\neq\emptyset$ so $f^{-1}(f(G))\cap f^{-1}(D)\neq\emptyset$, then $G\cap f^{-1}(D)\subseteq f^{-1}(f(G))\cap f^{-1}(D)$. So how can I guarantee that $G\cap f^{-1}(D)\neq\emptyset$ or any solution for that?
Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.  I am studying Baire space and I would like to try the following facts :      $(i)$ The inverse image of dense set is dense.       $(ii)$ The inverse image of comeager set is comeager.   I am studying Baire space and I would like to try the following facts  Can we collaborate to verify the above claims?    Thanks.
How to set the default program for opening files without an extension in Windows? How to set the default program for opening files without an extension in Windows? I am trying to solve the following two problems:  1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers.  2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and find all $z \in \mathbb C$: $\sin(z) \in \mathbb R$.  For 1),I've tried to prove it for the function $\cos(z)$ (I suppose the other one is analogue)  so I've used the fact that $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$.  Let $w \in \mathbb C$, I want to show there exists $z \in \mathbb C : f(z)=w$, i.e., $\dfrac{e^{iz}+e^{-iz}}{2}=w$, multiplying by $2$ and then by $e^{iz}$ yields $e^{2iz}+1=2we^{iz}$ iff $e^{2iz}-2we^{iz}+1=0$.   I don't know if this approach is the correct one but here I've replaced $e^{iz}$ by $x$, so the solutions of the equation would be the roots of the polynomial $p(x)=x^2-2wx+1$, by the quadratic formula, I get that $x \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$, where $w_0^2=4w^2-4$.  Then, $e^{iz} \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$. At this point I got lost, I would like to explicitly show that $z$ exists and I can't see existence directly from the fact that $e^{iz}=\dfrac{2w+w_0}{2}$ or $e^{iz}=\dfrac{2w-w_0}{2}$.  First I thought of taking logarithm of both sides of the equation ir order to solve for $z$, but this is not a legitimate operation unless $e^{iz} \in \mathbb R$.   I couldn't go any farther.  For point 2) I have no idea what to do, should I use the identity $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$?  I would appreciate some help with the two points (specially with point 2), at least in 1) I could do something).   Btw, Happy new year! Origin of the dot and cross product? Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it. I'm looking for a high current DC switch (able to switch something like 5V@30A) and I've encountered several DPDT that might do what I need, however they are only rated for AC applications (3A@250VAC or 6A@125VAC). Are these rating convertible to DC? or is that something that only the facturarer would know? Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$?  Edit: (t.b.) See also  for further answers. Why are images formed? How do rays of light getting focused at a point create an image?  How are the information of object's shape and dimensions expressed as an image when light rays converge?  Is it understandable by solving wave equation for light? Why does the SARS-CoV2 virus not remain infectious forever? Or does it? Given that the majority of biologists do not currently consider viruses to be alive, a virus can never die.  It can, however, get destroyed by long exposures to soapy water, alcohol, and apparently certain frequencies of UV light.  Why is an individual SARS-CoV2 virion not infectious forever?  Or is it? Uncountable chain of subsets of $\mathbb N$ How does new shield system work? Comma as decimal marker with siunitx (Spanish usage) I'm using siunitx to input some big numbers and units in my document.  Is there a way to make the output of \num{3.14} be "3,13" instead of "3.13"? In Spanish, we use the first one. Is there a simple way to explain how Kepler's third law follows from the inverse square law that of gravity  (and laws of motion)  For example for Kepler's second law we can say it's because Gravity is a central force  What is it about the law of gravity which ultimately gives rise to Kepler's third law? Explanation of terminal objects using limits I understand terminal objects, but I falter when trying to achieve them using the approach of diagrams, the category of constant diagrams $\mathcal{C}^{\mathcal{J}}$ and limits.  I have the empty category 0, now I have a diagram $D: \mathcal{J} \to \mathcal{C}$. However this diagrams image should be empty, because there are no objects or morphisms to map. At this point, I'm confused, because I can't get to the universal property $\forall c \in \mathcal{C}. \exists! f: c \to T$ anymore, because I'm missing T. (EDIT: I get that this is how it is supposed to be)  But continuing with the category of constant diagrams, even if I have a constant diagram mapping everything to one object / morphism, there is nothing to map, so the functors image is still empty. So now I'm missing both $T$, and any kind of $c$.  EDIT: It seems to me, have I chosen 1, instead of 0, I would have come up with that is considered the definition of the terminal category. What am I missing? Find the probability of an event determined by an inequality on exponential random variables Let X and Y be independent random variables such that $X∼Exp(1)$ and $Y∼Exp(2)$.  Find the probability that $3X+4Y≤5$.  I thought the equation for solving this was the integral from $\int_0^a \int_ 0^{(a-y)} \lambda_1 e^{-\lambda_1 x}\lambda_2 e^{-\lambda_2 y} dx dy$ where $a = 5$, so I first started with the integral from $\int_0^5 \int_0^{(5-y)} 3 \times 1e^{-1x} \times 4 \times 2e^{-2y} dx dy$ and got $23.83828927$ which is much larger than a probability can be. Any help? When can you switch the order of limits? Google Maps API v3, jQuery UI Tabs, map not resizing Plot showing discontinuity where it shouldn't How can I disable the 2nd hard drive in my 27" iMac? Work out the type of c# application a class contained in a DLL is being used by Since I updated Windows to 8.1 my Grub menu disappeared. I ran Ubuntu using Live USB, I used Boot-Repair but something went wrong. If I click on Reccomended repair I get this error:     GPT detected. Please create a BIOS-Boot partition (>1MB, unformatted filesystem, bios_grub flag). This can be performed via tools such as Gparted. Then try again. Alternatively, you can retry after activating the [Separate /boot/efi partition:] option.   So I tried to set my sda9 (I think that it's the Ubuntu system partition, since it's ext4) with the bios_grub flag and then to run Boot-Repair again. It didn't work.  I don't know what to do. My Grub was working fine (using UEFI) before I updated Windows.  I removed the bios_grub flag. Then I used fsck and it didn't find any error. So I tried again with Boot-Repair but it always gives me the "GPT detected" error. My boot is on UEFI mode, Secure Boot is "off".  When I boot my PC with SecureBoot OFF, I can reach the GRUB menu. But it's empty (the only voice is "System Setup").  I mounted my Ubuntu system disk:  sudo mount /dev/sda9 /mnt   and the other things:  sudo mount --bind /dev /mnt/dev sudo mount --bind /proc /mnt/proc sudo mount --bind /sys /mnt/sys   I logged on the system with chroot:  sudo chroot /mnt   I re-installed grub:  grub-install /dev/sda update-grub2   But it still doesn't work. I wrote a DLL in .NET and I want to access it in VBScript. I don't want to add it to the assembly directory.   Is there a way to point too the DLL and create an instance of it?