 $\tau$, and so closed, or
$B = \bigcup_{n=1}^{\infty} B_n$ where each $B_n$ is open, and so open, or 
Some trivial case in which $B$ is obviously open/closed, or
$B = \bigcap_{n=1}^{\infty} B_n$ where these are a mix of closed and open sets. Even if, ideally, all of the $B_n$'s were open, this does not necessarily give an open set, because only finite intersections yield open sets.

Thus, my conclusion is no. But could you construct some counter example for this? Thank you.
I encountered this confusion when trying to prove that every continuous function is Borel-measurable. I wanted to separate the proof into two cases (open, closed), but then I realized this. 

A:

(1) You are right in believing that the answer is no. For example, in $\mathbf R$, we have the Borel set
$$ B = (0,1] = \bigcap_n \left(0,\frac{n+1}n\right) $$
which is neither open nor closed.
(2) The possibilities that you list, are not all. When we denote set of the open sets by $B_0(\def\R{\mathbf R}\R)$ and define inductively
$$ B_n(\R) = \left\{\bigcap_k A_k\mid A_k \in B_{n-1}(\R) \text{ or } \R\setminus A_k \in B_{n-1}(\R) \right\} $$
then $B_{n-1}(\R) \subsetneq B_n(\R)$ for each $n$, and $\bigcup_n B_n(\R) \ne \textsf{Bor}(\R)$ (you have to continue the induction until $\omega_1$).

 Q:

Control order of script executions in test_package

Is it possible to control the order in which test_package executes testing scripts? I would like to start by executing some code that creates some objects shared by all individual tests. Therefore, this script needs to be executed first, before the actual test-blabla.R scripts.
I could try something like test-AAA.R or test-000.R, but I am not sure if the dir
$ \tau$, and so closed, or 
 $ B = \bigcup_{n=1}^{\infty } B_n$ where each $ B_n$ is open, and therefore loose, or 
 Some trivial case in which $ B$ is obviously open / shut, or 
 $ B = \bigcap_{n=1}^{\infty } B_n$ where these are a mix of closed and open set. Even if, ideally, all of the $ B_n$'s were open, this does not necessarily move over an open set, because entirely finite intersections yield capable set. 

 Thus, my conclusion is no. But could you construct some counter example for this? Thank you. 
 I meet this confusion when trying to prove that every continuous function is Borel - measurable. I want to separate the proof into two cases (open, shut), but then I realized this. 

 adenine: 

 (1) You are right in believing that the answer is no. For case, in $ \mathbf R$, we have the Borel set 
 $ $ B = (0,1 ] = \bigcap_n \left(0,\frac{n+1}n\right) $ $ 
 which is neither open nor closed. 
 (2) The possibilities that you list, are not all. When we denote set of the open sets by $ B_0(\def\R{\mathbf R}\R)$ and define inductively 
 $ $ B_n(\R) = \left\{\bigcap_k A_k\mid A_k \in B_{n-1}(\R) \text { or } \R\setminus A_k \in B_{n-1}(\R) \right\ } $ $ 
 then $ B_{n-1}(\R) \subsetneq B_n(\R)$ for each $ n$, and $ \bigcup_n B_n(\R) \ne \textsf{Bor}(\R)$ (you have to continue the induction until $ \omega_1 $). 

  Q: 

 Control order of script execution in test_package 

 Is it possible to control the decree in which test_package executes examination scripts? I would like to start by executing some code that creates some objects shared by all individual tests. consequently, this script needs to be executed first, before the actual test - blabla. R scripts. 
 I could try something like test - AAA.R or test-000.R, but I am not sure if the dir
 $\tak$, and so closed, or
$B = \biggup_{n=1}^{\infty} B_n$ whete each $U_n$ is olen, and ro open, or 
Some trivial case ib whixh $B$ is obviously open/zlosed, or
$H = \bigcap_{n=1}^{\inhty} B_n$ where these are a mix or clovxd and open setx. Even if, hdeally, all of tfe $B_n$'s were open, this does not necesswrily govf an open set, fecalsq onmj ninite intersections yield open aets.

Thuv, my conclusipn is no. But could you condtruft some counter exwmple for tyis? Eyank you.
I enzountered uhns confusioh when trying to prove that evefy coutinuous fubcrioj is Borel-meesurabje. I wanted bp sepasate thr proof into teo rasew (open, closed), but thei I realized this. 

A:

(1) Iou are richc in believing that tye answgr is no. Wir dxajpke, in $\mahhbh R$, we have the Borel wet
$$ B = (0,1] = \bigcap_n \legt(0,\sgsc{n+1}n\right) $$
whjch is nqither open nor closed.
(2) The possibilitiev tgat you list, are not alo. When we denote set lf the opqn sets by $B_0(\def\R{\mathbf R}\R)$ and define inductively
$$ T_n(\R) = \mdft\{\ylncap_y A_n\mid A_k \in B_{n-1}(\R) \text{ or } \R\setminus A_k \in B_{n-1}(\R) \ryfhu\} $$
tmen $B_{n-1}(\R) \subsetnez B_n(\R)$ for escj $m$, and $\bigcup_n C_n(\R) \ne \testsf{Bor}(\R)$ (you have ho contynue rhe inducuion intil $\omega_1$).

 Q:

Control order if script exvcutuons in test_packagz

Is it possiyle to conttol the order in which cest_padkage execuhes testihe scripts? I woula lpke do start by executing some code thav crectes somd obkects fhared by wll ikgividual tests. Thegeforg, this script nefds to be executed first, before the actual text-tlatla.R scrnpts.
I gould try sometring like test-CAA.R or cest-000.R, cut I am nkt sure if the dir
$\tau$, and so closed, or $B = where $B_n$ is and so open, which is obviously open/closed, $B = \bigcap_{n=1}^{\infty} where these are a mix of and open sets. Even if, ideally, all of the $B_n$'s were open, this not necessarily give an open set, because only finite intersections yield open sets. my is But you construct some counter example for this? Thank you. I encountered this confusion when trying to that every continuous function is Borel-measurable. I wanted separate the proof into cases (open, closed), but then realized A: (1) are in that the answer no. For example, in $\mathbf R$, we have the Borel set $$ B = (0,1] = \bigcap_n $$ which open nor (2) possibilities you list, are When we denote set of the $B_0(\def\R{\mathbf R}\R)$ and define inductively $$ B_n(\R) = A_k\mid A_k B_{n-1}(\R) \text{ or } \R\setminus A_k B_{n-1}(\R) \right\} $$ then $B_{n-1}(\R) \subsetneq B_n(\R)$ for $n$, and $\bigcup_n B_n(\R) \ne \textsf{Bor}(\R)$ (you have to continue the induction until $\omega_1$). Q: of script executions in Is it possible control order which executes testing I would like to start by executing some code that creates objects shared by all individual tests. Therefore, this script needs executed before the actual scripts. I could try like or test-000.R, but I sure the
 $\tau$, and so closed, or
$B = \bigcup_{n=1}^{\iNfty} B_n$ wherE each $b_n$ iS opEn, And sO opeN, or 
Some trivial CAse iN which $B$ is obviously open/CloseD, oR
$b = \bigCAp_{N=1}^{\inftY} B_n$ wherE ThESE arE a MiX of ClOSeD and oPen Sets. EveN if, ideally, All Of The $B_n$'s were opEN, tHis does not NecEssarily give An oPen set, BeCauSE only FinIte inTersecTIons yiEld open seTs.

tHus, my cONclusioN IS nO. But Could you construct SOmE Counter example For thiS? THAnK YOu.
I EncOuntered thIs ConfuSIon when TRyING To pROve that every cOntinuous fuNCtiOn is BoReL-meASurablE. I wanTeD To sEparate the pRoof Into two caSes (opeN, Closed), bUT then I rEalizeD thIs. 

A:

(1) you aRE rIgHt iN bELieVInG thAT thE answer iS nO. FOr exaMple, IN $\MAThbf r$, we Have The BoRel set
$$ B = (0,1] = \bigcap_N \leFt(0,\frAC{n+1}n\Right) $$
Which Is neItHer opEn nor cLosed.
(2) thE possibilities tHat yOu list, are Not AlL. WhEn We denOTe set oF thE opEn sets bY $B_0(\def\R{\mAThbF R}\r)$ AND dEfine inductively
$$ B_n(\r) = \lEFT\{\bIgcap_k A_k\Mid A_k \iN b_{n-1}(\r) \tEXt{ or } \R\setMiNus a_k \in b_{N-1}(\r) \righT\} $$
theN $b_{n-1}(\r) \subsetnEq B_n(\R)$ fOR eAcH $n$, and $\biGcUp_n B_n(\R) \Ne \TexTsf{bor}(\R)$ (yOU havE to conTinue the InducTIon until $\omega_1$).

 Q:

cOntrol order of SCrIPT eXEcutIonS in test_packAge

IS It poSsibLE tO coNTrol tHe ordEr IN wHIch test_package execuTeS testiNg scrIpts? I would likE to start by EXECuting soMe coDE tHAt creates some oBjectS shared by aLL individUal teSts. ThereFore, this sCRIpt needs To bE exEcuTed FIRsT, before the actUAL tesT-bLabla.R sCriPts.
I couLd tRy sOmeThiNg Like test-AaA.R or tesT-000.R, BuT I Am Not Sure iF The dir
 $\tau$, and so closed, or 
$B = \big cup_{ n=1 }^{ \i nfty } B_ n$ where each$ B_n$  is open, and so open,  or So m e tr i vi al ca se in w h ic h $B$  i sobv io u sl y ope n/c losed,or
$B = \b igc ap _{n=1}^{\inf t y}  B_n$ wher e t hese are a m ixof clo se d a n d ope n s ets.Even i f , idea lly, allof the $B _ n$'s we r e  o pen,  this does not ne c es s arily give anopen s et ,  b e c aus e o nly finite  i nters e ctionsy ie l d ope n  sets.

Thus,  my conclus i onis no.  B utc ould y ou co ns t ruc t some coun terexample f or thi s ? Thank you.
Iencoun ter edthis co nf usi on whe n  t ryi n g t o proveth at  ever y co n t i n uous  fu ncti on is  Borel-measur abl e. I wan ted t o sep arat ethe p roof i nto t wo  cases (open, c lose d), but t hen  I  re al izedt his. 
A: 

( 1) Youare rig h t i nb e l ie ving that the answ er i sno. Forexampl e ,in $\mathbf  R $,we h a v e the  Bor e lset
$$ B  = (0, 1 ]=\bigcap _n  \left (0 ,\f rac {n+1} n \rig ht) $$ 
which i s nei t her open nor c l osed.
(2) The po s s ib i liti esthat you li st,a re n ot a l l.  Wh e n wedenot es et of the open sets by  $ B_0(\d ef\R{ \mathbf R}\R) $ and defi n e inductiv ely$ $B _n(\R) = \left \{\bi gcap_k A_k \ mid A_k\in B _{n-1}(\ R) \text{ o r } \R\s etm inu s A _k\ i nB_{n-1}(\R) \ r i ght\ }$$
then  $B _{n-1}( \R)  \s ubs etn eq  B_n(\R)$  for eac h$n $,  a nd$\big c up_n B_n (\ R)\n e \ texts f {Bor}( \R)$(you  h av e  to  contin u et h e in du ct ionunt il  $\om ega_ 1 $). 

 Q:
Control o rde r  ofsc ri pt exec utions in tes t_ package

I sitpossib l e  to cont rol the order in whicht est_pac kag e exe cute s testing  sc ripts?  Iw ould l ike to  star tbye x ecuti n g  s ome  c ode that c r e ate s som eobje cts sha red by all individ u altests. Theref ore , th i s  s cri p tn eed st o b e executed first,  before th ea ct ual test-b l abl a. R scrip ts.
I c ouldt ry some thing lik e test-AA A. R or t est -000.R, bu t I am n ot sure i f  thed ir 
 $\tau$,_and so_closed, or
$B = \bigcup_{n=1}^{\infty}_B_n$ where_each_$B_n$ is_open,_and so open,_or 
Some trivial_case in which $B$_is obviously open/closed,_or
$B_= \bigcap_{n=1}^{\infty} B_n$ where these are a mix of closed and open sets. Even_if,_ideally, all_of_the_$B_n$'s were open, this does_not necessarily give an open_set, because_only finite intersections yield open sets.

Thus, my conclusion_is_no. But could_you construct some counter example for this? Thank you.
I_encountered this confusion when trying to_prove that every_continuous_function_is Borel-measurable. I wanted_to separate the proof into two_cases (open, closed), but then I_realized this. 

A:

(1) You are right in_believing that the answer is no._For example, in $\mathbf R$,_we have_the Borel set
$$ B =_(0,1] = \bigcap_n_\left(0,\frac{n+1}n\right) $$
which_is neither open_nor closed.
(2) The possibilities that you_list, are not_all. When we denote set of_the_open sets by_$B_0(\def\R{\mathbf_R}\R)$_and define_inductively
$$ B_n(\R) =_\left\{\bigcap_k_A_k\mid A_k_\in_B_{n-1}(\R) \text{ or } \R\setminus A_k_\in_B_{n-1}(\R) \right\} $$
then $B_{n-1}(\R) \subsetneq B_n(\R)$ for_each $n$, and $\bigcup_n_B_n(\R)_\ne \textsf{Bor}(\R)$ (you have_to continue the induction until_$\omega_1$).

 Q:

Control order of script executions_in test_package

Is_it possible_to control the order in which test_package executes testing scripts? I_would like to start by executing_some code that creates_some objects_shared_by all individual_tests._Therefore, this_script needs to be executed first, before_the actual_test-blabla.R scripts.
I could try something like_test-AAA.R or test-000.R, but_I_am not sure if the dir