I have an exercise in general topology as follows:  Let $f:X\rightarrow Y$ be a map between topological spaces. Assume that $X=\cup _{i\in I} X_i$ with $X_i$ closed subsets of $X$, and $f|_{X_i}: X_i\rightarrow Y$ is continuous for every $i$. We can prove that if $I$ is finite, then $f$ is continuous on $X$. However, do we have a counter-example when $I$ is countable?  Thanks for your help!
The gluing lemma for closed sets states:     Given a finite closed cover $\{A_i\}$ of a topological space $X$, together with continuous maps $\{f_i : A_i \to Y\}$ into some other topological space $Y$, there exists a unique continuous map $f : X \to Y$ whose restriction to each $A_i$ is equal to $f_i$.   Question: What is a good and simple counter-example when the gluing lemma fails in the case that $\{A_i\}$ is infinite, but countably so.  My attempt: I have only been able find a silly counter-example: let $\{A_\alpha\}$ be the collection of all points of $X = [0,1] \subset \mathbb{R}$, and let $f_\alpha = 0$ for all $\alpha$ except for $\alpha = \alpha_0$, for which $A_{\alpha_0} = \{0\}$ and $f_{\alpha_0} = 1$.
There's a web services I want to call in my application, I can use it with importing the WSDL or by just use "HTTP GET" with the URL and parameters, so I prefer the later because it's simple thing.  I know I can use indy idhttp.get, to do the job, but this is very simple thing and I don't want to add complex indy code to my application.  UPDATE: sorry if I was not clear, I meant by "not to add complex indy code", that I don't want add indy components for just this simple task, and prefer more lighter way for that. Favourites Page on iOS App I would like to have a page listing all the questions I marked as favourite, same as the .  Also, I think the desktop version, the star, is much more intuitive and easier to find. Thanks! How can I understand a Vortex Tube and its efficiency? A  takes a pressurized input stream, most typically of a gas, and creates two output streams with a temperature differential. .  Both linked sources are scarce with information about how and why this works. Now, I have two questions:   Why does it work, specifically why should the situation in the vortex lead to a transfer of thermal energy from the inner stream to the outer one? How efficient can it be?  How do you define the efficiency of a device that may be closer to Maxwell's Demon than to a heat pump? My feeling is that any analysis should not only take into account the sum of input energy (thermal and mechanical energy instream) and sum output (thermal energies and pressures of both gas streams), but also the Temperature differential that is created - since that contains an ability to create work.* If course it's pointless to create heat from high grade 8mech.) energy to transform it back to mech. energy - but it gives an idea on the worth of the output. just after the Big Bang and before inflation the currently observable universe was the size of a coin. One millionth of a second later the universe was the size of the Solar System, which is an expansion much much faster than speed of light. Can space expand with unlimited speed? How do I check if an integer is even or odd? How can I check if a given number is even or odd in C? How can I download this flash game for offline play? is one of the best flash games ever.    Is there some way to download it, so that I can play it offline? So in the image below there are two circles, how can I remove the internal vertices between them without messing up the outside shape. I'm working on a gun and this is just a simple way to illustrate the issue I'm dealing with. If I where to go in and delete the internal faces it would leave holes on the outside of the object, is there a fast way to get around this? Science fiction short story: beam transporting people into the distant future I want to package my project in a single executable JAR for distribution.  How can I make a Maven project package all dependency JARs into my output JAR? My friend Evan,is griefing me. When I went on his world to get revenge, he banned me. The screen said " HerobineYoung has banned enderpizza you will be kicked out in 5 minutes." I begged him to tell me how but he wouldn't tell me.   How can I ban people in Minecraft PE? How to solve $x^3=-1$? How to solve $x^3=-1$? I got following:  $x^3=-1$   $x=(-1)^{\frac{1}{3}}$  $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$... The integral $\int_0^8 \sqrt{x^4+4x^2}\,dx$ How can I speed up D&D 3.5e combat? How can I migrate a Windows 10 from BIOS/MBR boot to UEFI/GPT without reinstalling? Performance of Non Clustered Indexes on Heaps vs Clustered Indexes compares the performance for individual select/insert/delete/update and range select statements on a table organized as a clustered index vs that on a table organized as a heap with a non clustered index on the same key columns as the CI table.  Generally the clustered index option performed better in the tests as there is only one structure to maintain and because there is no need for bookmark lookups.  One potentially interesting case not covered by the paper would have been a comparison between a non clustered index on a heap vs a non clustered index on a clustered index. In that instance I would have expected the heap might even perform better as once at the NCI leaf level SQL Server has a RID to follow directly rather than needing to traverse the clustered index.   Is anyone aware of similar formal testing that has been carried out in this area and if so what were the results? Is it correct to sometimes capitalize words mid-sentence in computer user interfaces? Metric on Cartesian Product I'm currently stuck on an exercise where I need to show that if $(M,d)$ and $(N,\rho)$ are metric spaces, then $d_2((a,x),(b,y)) := \sqrt{d(a,b)^2 + \rho(x,y)^2}$ defines a metric on $M\times N$. I've already shown that the basic criteria of a metric are met, but I still need to show that the triangle ineqaulity holds.  I've tried using the triangle inequality on $d$ and $\rho$ and estimating the square root with $\sqrt{a^2+b^2}\leq a+b$, but regardless of which order I've used them in, I seem to overestimate my target. I thought maybe I somehow need to use Cauchy-Schwarz, but I don't know how. Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$. Did not receive site association bonus (+100 rep) Why does the air flow faster over the top of an airfoil? I understand the common explanation of lift, which describes the airflow over the top of the wing as moving faster than the air below the wing.  However, I don't quite understand why the air moves faster.  I've read some explanations citing a circulation effect among others, but haven't found a good, clear explanation for the exact cause.  Can someone help me out?