$f:A \to B$ a homeomorphic mapping from $A$ to $B$  $=&gt; f:A \backslash U\to B \backslash f(U)$ is also homeomorphic, $U \subseteq A$.  Why does this implication hold?
Let $f:X \to Y$ be a homeomorphism between topological spaces $X$ and $Y$. Let $A \subseteq X$ and $B = f(A) \subseteq Y$ (given the subspace topology) and let $g: A \to B$ be the restriction of $f$ given by $g(a) = f(a)$ for $a \in A$.  Showing that $g$ is a homeomorphism.   $g$ injective   Let $g(a_1) = g(a_2)$, $a_1, a_2 \in A$.  $\implies f(a_1) = f(a_2)$  $\implies a_1 = a_2$ as $f$ homeomorphic.   g surjective   Let $y \in B$  $\implies y = f(a)$ for some $a \in A$.  $\implies y = g(a)$ for some $a \in A$.   $g$ continuous   Let $U \subseteq B$ be an open set in $B$.  $f(a) = g(a)$ for all $a \in A =&gt; f^{-1}(b) = g^{-1}(b)$ for all $b \in B$.  So $g^{-1}(U) = f^{-1}(U)$ which is open in $A$ as $f$ homeomorphic.   $g^{-1}$ continuous   Let $U \subseteq A$ be an open set in $A$.  $g(U) = f(U)$ which is open in $B$ as $f$ homeomorphic.  So $g$ is homeomorphic.  Is my understanding correct, this seems almost too straightforward so I was wondering have I overlooked something?
Fighting Spam - What can I do as an: Email Administrator, Domain Owner, or User? This is a  about Fighting Spam.   Also related:                    There are so many techniques and so much to know about fighting SPAM. What widely used techniques and technologies are available to Administrator, Domain Owners, and End Users to help keep the junk out of our inboxes?  We're looking for an answer that covers different tech from various angles. The accepted answer should include a variety of technologies (eg SPF/SenderID, DomainKeys/DKIM, Graylisting, DNS RBLs, Reputation Services, Filtering Software [SpamAssassin, etc]); best practices (eg mail on Port 25 should never be allowed to relay, Port 587 should be used; etc), terminology (eg, Open Relay, Backscatter, MSA/MTA/MUA, Spam/Ham), and possibly other techniques. "Open With" dialog ignores my selection Why is $M\frac{dv}{dt} = v_{rel} .\frac{dm}{dt}$ correct and $(M - dm)\frac{dv}{dt} = v_{rel} .\frac{dm}{dt}$ wrong? Newton's 2nd law of motion can't be applied for mass-varying systems. Another force, known as Thrust must come to play. It can be measured using law of conservation of linear momentum. $$\text{Thrust} M.\dfrac{dv}{dt} = v_{rel} .\dfrac{dm}{dt}.$$   where $M$ is the mass of the system;   $\frac{dm}{dt}$ is the rate at which the mass changes;    &amp;    $v_{rel}$ is the relative velocity of the exhausted mass w.r.t to the main system.    This is what my book writes. Now, the left hand side of the eqn. represents force on mass $M$ whereas the right hand side represents the force due to variable mass. How can they be equal? During the application of thrust, $M$ does change and the equation had to be $$ (M - dm). \dfrac{dv}{dt} = v_{rel} .\dfrac{dm}{dt}$$ But that is not the book's equation. Why is mine wrong? What is the intuition? I've noticed in the last couple of days that there have been a few occasions when there's been a sudden surge of questions in the SO "Reopen" review queue.  Normally there's either no questions at all to reopen or at most three or four, and that's still the case, except that for the last couple of days I've gone to the review page and seen about fifty-odd questions in the queue, where just moments before there had been none.  I've gone through the reviews and the questions themselves seem reasonably legitimate (including the occasional audit), so I'm not questioning that... it just seems odd that there would be so many more than usual all of a sudden, and that they'd all appear at once.  Anyone got any insights, or should I just chalk this down as "one of those things"? Short story or novel from the 1970s or before about two warring alien tribes Two Circles and Tangents from Their Centers Problem Let $\Gamma_1$ and $\Gamma_2$ be two non overlapping circles with centers $O_1$ and $O_2$ respectively. From $O_1$, draw the two tangents to $\Gamma_2$ and let them intersect $\Gamma_1$ at points $A$ and $B$. Similarly, from $O_2$, draw the two tangents to $\Gamma_1$ and let them intersect $\Gamma_2$ at points $C$ and $D$.  Prove that $AB=CD$.  I've done some extensive angle chasing on this but have been unable to make any real progress. Can't decide whether $ABCD$ is supposed to be rectangle (as in my diagrams), a parallelogram or even a trapezium. There is a homothety taking one circle to the other but as far as I can see this doesn't help as we don't have a clearly defined center to this.  Any help/hints would be greatly appreciated. Can't run a shell script So I have this script:  #!/bin/bash # Flips the screen (hopefully)  syntax_error=0 orientation=0  current_orientation="$(xrandr -q --verbose | grep 'connected' | egrep -o  '\) (normal|left|inverted|right) \(' | egrep -o '(normal|left|inverted|right)')" case $current_orientation in         normal)                 current_orientation=0         ;;         left)                 current_orientation=1         ;;         inverted)                 current_orientation=2         ;;         right)                 current_orientation=3         ;; esac  if [ $current_orientation -eq 0 ]; then         orientation=2 fi  if [ $current_orientation -eq 2 ]; then         orientation=0 fi method=evdev  # LENOVO S10-3t CHANGE ==&gt; Hard Coded my device number to 11!!!!!!!!  device=11  swap=0 invert_x=0 invert_y=0 real_topx=0 real_topy=0 real_bottomx=4020 real_bottomy=4020  case $orientation in         0)                 swap=0                 invert_x=0                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         1)                 swap=1                 invert_x=1                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;;         2 )                 swap=0                 invert_x=1                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         3 )                 swap=1                 invert_x=0                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;; esac  if [ $method = "evdev" ]; then         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axis Inversion" $invert_x $invert_y         xinput set-prop "$device" "Evdev Axis Calibration" $topx $bottomx $topy $bottomy         if [ $orientation = 2 ]; then                            xrandr -o inverted         fi         if [ $orientation = 0 ]; then                 xrandr -o normal         fi fi  #   It's for flipping screen in my Lenovo S10-3t. I copied it from the netbook's wiki page, and added the #!/bin/bash at the top. The filename is flipscreen.sh. How can I make it work? At what point is a creature's toughness calculated? If I have  where I can sacrifice a creature and put X 1/1 token creatures onto the battlefield, where X is the sacrificed creature's toughness.  If I have a creature that is 4/4 and cast  to give it +3/+3 then sacrifice it? Do I get 4 tokens or 7?  Also what if my 4/4 creature was dealt 2 combat damage before I sacrificed it? Do I get 2 or 4 creatures?  Finally, what about counters? If my 4/4 had a +1/+1 do I get 4 or 5 tokens? Why does my server show              total      used      free   shared buffers    cached     Mem: 12286456  11715372    571084        0   81912   6545228 -/+ buffers/cache:  5088232   7198224    Swap: 24571408     54528  24516880   I have no idea on calculating the memory in linux. I think it says that 5088232 is used where as 7198224 is free, meaning it is actually consuming 5GB of RAM? Can I force WordPress to use  instead of ? Remove default ID attribute from HTML helpers in ASP.NET MVC Comparing complex numbers If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not?   Can somebody give explanation on this.... I will be grateful to him. Why *not* parse `ls` (and what to do instead)? What kind of noun is 'picture'? I'm not sure of the right place to ask this, but I got confused trying to understand how the computer will interpret the sentence:     This is my picture.   In actual sense, the real owner of the picture should be the photographer not the person who was photographed... or am I getting something wrong? So, it seems the confusing thing about 'picture' is determining who the real owner of the picture is. So is there a special category for these kind of nouns or an official clarification somewhere?  Thanks When replacing non-grounding type receptacles is a GFCI receptacle required, or can any GFCI device be used? Under the NEC (it was 210(7)(d)(3), but appears to have moved around some), it is permissible to replace an ungrounded receptacle with a GFCI receptacle, then wire further ungrounded or non-grounded (i.e. a NEMA 5 without a ground wire hooked up) receptacles on the circuit to the load side of the GFCI, all in lieu of running a ground wire in order to replace the existing ungrounded receptacles with grounded receptacles.  However: does this permission extend to replacing the branch circuit overcurrent protection device with a GFCI-type circuit breaker (or a CAFCI breaker that provides ground fault protection for personnel)?  Or can only GFCI receptacles be used for this retrofit? The NEC's verbiage does not make mention of breaker-type GFCIs... Find html element which id starts with Not to Mention ≈ [Let Alone ≈ Much Less ≈ Still Less] For brevity, I symbolise synonymity with ≈.  So  X ≈ Y  means  X and Y are synonyms.  From : let alone ≈ not to mention  From :  let alone ≈ much less ≈ still less.  Are these all synonyms? What are the similarities and differences between each? Finding the nth non square free number We all know what  numbers are. I was more interested in checking an efficient way to find nth non square free number. Could cancer be in itself a evolutionary process? Could cancer be in itself a evolutionary process? Maybe in some way could it be a process of variation? Or would this idea be completely without support, if so, why?   I don't mean that each case would lead to evolution, but that within an entire group of organisms, it could lead to case of of individuals developing ways to resist what caused the cancer, maybe in cases of individuals beating the cancer or in cases of successive generations of off spring from individuals with cancer?   Could it be possible that in rare cases it could lead to the development of new organs or specialized cells? I gave a quick answer to part of  which had not been covered by previous answers, trying to clarify the reason you would say time of decoding but not decoding’s time. I said it was ’s usually indicates possession, but of course there were several counterexamples that would have occurred to me after a moment’s consideration, and these where helpfully supplied:   Britain’s climate two days’ time  a day’s work  the sun’s rays   I am still of a mind to say that possession of some sort is what allows the ’s. Even though the sun does not have title to its rays, they do belong to the sun. Now, at the risk of duplicating the original question and/or being pigheaded, I am curious as to why time of decoding but not decoding’s time is correct, if not for the reason I gave.