Difference between $\mathbb R^m\times\mathbb R^n$ and $\mathbb R^m\oplus\mathbb R^n$ I'm reading an analysis book and sometimes the author uses several denotations: $\mathbb R^{m+n}$, $\mathbb R^m\times\mathbb R^n$ or $\mathbb R^m\oplus\mathbb R^n$(for me they are essentially the same thing).  The first two spaces are homeomorphic to each other, I think when he uses $\mathbb R^m\times\mathbb R^n$ he want to emphasize  some open subsets $U\subset \mathbb R^m$ and $V\subset \mathbb R^n$. However there is another problem, these spaces are diffeomorphic? if don't, we can't say they are essentially the same in the context of real analysis.  If my reasoning is right so fat, What I really don't understand is why he uses the direct sum $\mathbb R^m\oplus\mathbb R^n$. Does he want to emphasize the underline operation between the two spaces? this is not clear in the book.   Am I right? what is the difference between  $\mathbb R^m\times\mathbb R^n$ and the external direct sum $\mathbb R^m\oplus\mathbb R^n$ in this context?
The direct sum $\oplus$ versus the cartesian product $\times$ In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times A$; however in, for example, the fundamental theorem of finitely generated abelian groups, we normally write that every such group is isomorphic to one of the form $$ \mathbb{Z}^n \oplus \mathbb{Z}_{r_1} \oplus \cdots \oplus \mathbb{Z}_{r_t}  $$ where $\mathbb{Z}^n$ now means $\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}$.  From an intuition perspective, and in the sense of sets, is this more or less the same as $\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$? (Bear in mind I am normally using these ideas in relation to homology groups.)
In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points... but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.  I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?  Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.  I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology? Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? Is it still bad to use swap on a modern SSD? I have a Samsung 960 Pro SSD with Ubuntu 18.04  is it safe to put the swap file on there or would it hurt it in the long run,  I heard some people saying that new SSDs don't suffer from that issue anymore, is this true? how implement the jqueryUI in visualforce Is there a robust command line tool for processing csv files? jQuery: Uncheck other checkbox on one checked Where do I find the power cells to restore the bunker door in “Ancient armory”? I am trying to find the power cells to access the amor. Yet the placement is unmarked and I thought I would eventually stumble over them clearing the cauldrons.  Yet after clearing three cauldrons (RHO, XI, and Sigma) I still only have the one from the very beginning.  Where do I find all the power cells? Rack Mount Servers - Build them or Buy them? This is the  for "Should I build coomputing hardware myself?" questions.   I have put together countless PCs, but never a large server. The geek in me says build it, but the realist in me says let the manufacturer handle it when there is a problem. Ignoring the time penalty involved with the initial assembly time of a built one, which is a better solution? Have you ever run into a problem with a home build server that would have been solved easier/quicker/cheaper by going with a manufacturer? Are there any features that manufacturers give that aren't easily attainable with a home built server? Order of "not" with infinitive We are having issues with our exchange 2013 server. The server is running Windows Server 2008 R2 Standard.  The issue we're having is that our Exchange server isn't receiving internet emails. It can currently send mail out to internet addresses, and send and receive internal emails from the current email server. I have checked the settings of the receive connectors and everything seems correct. We have tried everything that we can think of.   We can send and receive email locally, and send out to a gmail account. However, we don't get the reply email from the gmail account to the inbox.    Thanks in advance!! Image is of the DNS settings on the exchange server. Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. I want to prove that:  $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. I'd like to be able to decompose a concave mesh into a set of convex meshes for 2 reasons:   Transparent rendering Physics shapes   Is there an algorithm that takes a set of triangles (concave) as input and outputs a number of sets of triangles (convex)?  I'd like it to not fill in the holes between parts of the original mesh.  I've already come across a small idea: find all the concave edges, and split the meshes along the edge loops.  Am I on the right track?  How could I implement this? So  i have a scene with a few objects, and a water simulation in the scene. However I want to animate the camera turning around in the scene, while the water simulation is frozen in time. So something like a time freeze effect. But when ever I try to render an animation it renders the water simulation too. I just need one frame of the water simulation. Is there a way to do this? Number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$ Does anyone know of any linux tool specifically designed to treat files as sets and perform set operations on them? Like difference, intersection, etc? How to create a node with submissions of a webform? I'm using Drupal 7 and created a content type 'NewArticle' with the fields 'Title', 'Text' and 'Image'. Users have the permission to create a 'NewArticle', but I don't like the way of displaying the administration bar in order to let them create a 'NewArticle'.  Instead, I'd like to work with . My goal is to create a Webform with exactly the same fields as the content type 'NewArticle'. So what I want is: when users filled in the form and submit it, a new node 'NewArticle' is created with the submitted information.  How can I do this? Are there modules that make this possible? I can't find any... elementary row operation change the eigenvalue of a matrix The Software Center has a section of For Purchase programs &mdash;see screenshot below&mdash;, which is annonying me every time I want to install something.  Is there any way to remove or hide these programs from the Software Center?     Links to answers:                     Please leave a comment or send a mail to jmendeth@gmail.com   if it's not working for you or you're having trouble! How to call a space in which every G$_\kappa$-set is open Recall that in a topological space $X$, we say that a subset $A\subset X$ is a G$_\delta$-set if $A$ is the intersection of countably many open sets of $X$. Also, we say that $X$ is a  if G$_\delta$-sets are open.  Now, call as G$_\kappa$-set the intersection of $\kappa$ many open sets, where $\kappa\geq\aleph_0$ is a cardinal. I would like to know if there is some standard name for those spaces in which G$_\kappa$-sets are open, for a fixed cardinal $\kappa\geq \aleph_0.$  Motivation: this kind of space arose in my PhD thesis, but I do not know any references in which they appear. I'm temporary calling them as $P_\kappa$-spaces. Symmetrical twin paradox in a closed universe Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.  On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to , so that one frame is preferred - but which one?  So the question is: Who will be older? And why?  And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...  How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT?  And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know  is only a local theory but I don't understand why the global preferred frame should not 'override' the local one).  Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!)  EDIT Because there are still things that are unclear to me I posted a follow-up question: