If A is compact and B is closed then A Intersection B is compact. I tried to solve it using the fact that compact set is closed and bounded but the problem here they did not tell us in which topology is compact or if its compact in R.
Fact  The intersection of a closed set $A$ and a compact set $B$ is closed.  Background  I wanted to prove that the intersection of closed and compact sets is compact. There might be other ways, but I am not interested in them for now and am looking to prove the above statement.  Relevant Facts  I don't have much idea what to use. $A^c$ is open. Every cover $\cup_{i\ge1}U_i$ of $B$ contains a subcover $\cup_{i=1}^kU_i$. I might want to prove $A\cap B^c$ is open. Limit points may be another points to consider but I don't get any idea in that direction.  I also don't have that $B$ is closed otherwise it would be trivial.  Some Clarification  I am aware that compact sets are not closed in arbitrary topological spaces. The reason why I was motivated to believe tha intersection of closed and compact set is compact is true is: See Matt E.'s Answer      So if $K$ is compact in an arbitrary topological space (which is just to say that   it is a compact topological space when given its induced topology) and     $L$ is closed then $K \cap L$ is a closed subset of $K$ in its induced topology   Is it true if $A$ and $B$ are not whole spaces?  I would appreciate any hints.
Python: using a recursive algorithm as a generator Binding energy and mass I’ve been told that a greater binding energy means the nucleus is more tightly bound, and therefore that decreases the mass of the nucleus with respect to its nucleons when separated. But why does a tighter nucleus mean a smaller mass? Relative Pronouns are ALWAYS added right after antecedent? Is is a rule that relative clauses should be added right after the antecedent(the noun relative pronouns refer to)? What if it is confusing?   For example:  "I went for a spin with a handsome boy last night."  "The handsome boy went out with my friend."  "I went for a spin with a handsome boy who went out with my friend last night ."  "I went for a spin with a handsome boy last night who went out with my friend." The web is littered with any number of pages () giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no such pairs for primes congruent to 3 mod 4).  However, none of the stuff I've read on the topic offers any help with actually efficiently finding (ie other than a straight search up to sqrt(p)) the concrete values of such squares.  What's the best way to actually find them ? Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$ I need your help with evaluating this limit:  $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$  i.e. we apply the $\sin$ function $n$ times.  Thank you. Finite choice without AC Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here). Get list of all tables in Oracle? How do I query an Oracle database to display the names of all tables in it? Distribute objects optimaly So I've got a file with loads of simple objects. This will later be used to cut out these shapes out of plastic. So, using the material optimally is of outmost importance. I was wondering if there is any script or plugin that could automaticly arrange the objects within the artboard so they could fit best. Are \( and \) preferable to dollar signs for math mode? Along the lines of , what reasons are there (if any) to favor \( ... \) over $ ... $? Has 'Ubuntu Software' been renamed to 'Snap Store'? I updated Ubuntu last night (sudo apt get update &amp;&amp; sudo apt get upgrade) and to  my surprise this morning I didn't find the 'Ubuntu Software' app and there's a new entry called 'Snap Store' which when opened looks just like the old 'Ubuntu Software' app. So has 'Ubunu Software' been renamed to 'Snap Store'? And if yes, why? What are the minimum requirements to run Windows 10? If I did  \setcounter{chapter}{-1} \chapter{Introduction}   then I get Chapter 0 Introduction. I just want the word Introduction, and I want it also in the contents when I use   \tableofcontents   How can I do that? ColA                  Col B              ColC Project A            A.pdf               A.pdf Project B                                No Link   I have a list of project records. One column (e.g., B) may have a reference to a pdf file. If I create a formula   =if(isblank(B2),"No Link",hyperlink(folderURL + B2),B2)   What I want is to place a hyperlink on only those rows where column B contains a pdf reference, and insert text "No Link". I get the proper text values, but in all rows have the hyperlnk default appearance and behavior -- Excel seems to be creating "phantom links" out of the "No Link" cells.     Clicking on the "No Link" value in Col C activates an attempt to open the file listed in Col B, which returns a No file found error. How to convert int[] to Integer[] in Java? Do we keep hats from year to year? This is the first year that I pay attention to hats (I love hats!) and I wonder if I will keep my hats for next year?  Or will I start over with zero hats next year? Why does the Replace() string method not modify my string variable? Cheaper zoom lenses usually are faster at the wide end and slower at the long end (for example, the $150 Canon EF-S 18-55mm f/3.5-5.6). More expensive constant-aperture zoom lenses have the same aperture regardless (for example, $800 Canon EF 17-40mm f/4.0 L).  My question is: are these good lenses  at the wider settings, or do they have a different optic system that allows them to maintain the same aperture throughout the zoom range? Learning Regular Expressions Error message in my excel formula There's already a question like this  so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.   Is there a way to derive  from the Lorentz' Force law or just from Maxwell's Equations?   The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.   Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?  I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.   In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?