#noindex
##==클로저,closure =,closure 클로저 closure
번역들
 논리학에서
  ''(번역어를 대체하는 얘기 도중)'' '닫힌 식(closure)'을 '폐쇄식'으로 바꾸었다 ... (기호논리학 p10)
  MKPAGE [[닫힌식,closure]]? ... aka [[닫힌식,closed_expression]] ??
   is_a [[식,expression]] > [[논리식,logical_expression]]?
   or is_a [[포뮬러,formula]] > [[논리식,logical_formula]]?
----
[[토큰,token]]을 다루는, [[언어,language]]에 대한 [[연산,operation]]s들 중에서 (language operations)
via https://www.cs.fsu.edu/~engelen/courses/COP562105/ 
 * Union
  $L\cup M=\left\lbrace s \middle| s\in L \text{ or } s\in M \right\rbrace$
 * concatenation
  $LM=\lbrace xy | x\in L \text{ and } y \in M \rbrace$
 * exponentiation
  $L^0=\lbrace \varepsilon \rbrace$
  $L^i = L^{i-1}L$
 * '''Kleene closure'''
  $L^* = \bigcup_{i=0,\ldots,\infty} L^i$
 * '''Positive closure'''
  $L^+ = \bigcup_{i=1,\ldots,\infty} L^i$

Kleene_closure
Ggl:"Kleene closure"
positive_closure
Ggl:"Positive closure"
----
수학(algebra?)의 것은 [[폐포,closure]]로 번역됨, 이것에 대해
일단은 goto [[폐포,pyepo]]


NdEn:closure
Ndict:closure
WtEn:closure
WpSp:Closure
WpEn:Closure

= 관련표현 =
[[closed_]]
 [[Date(2023-09-08T21:15:05)]] 현재 다음하나뿐
 [[닫힌집합,closed_set]]
WtEn:closed ?  https://mathworld.wolfram.com/Closed.html

WtEn:closedness = https://en.wiktionary.org/wiki/closedness
 ([[Date(2023-11-22T15:55:50)]] 간단, "The property of being closed.")

opp. openness ? [[열림성,openness]]? WtEn:openness = https://en.wiktionary.org/wiki/openness (unimportant)

opp. [[open_]] ?

KmsE:closure

pagename 후보:
[[닫힘,closure]] =닫힘,closure =,closure 닫힘 closure
{
KmsE:closure

}
[[폐포,closure]] =폐포,closure =,closure 폐포 closure
{
[[WpKo:폐포_(수학)]] = https://ko.wikipedia.org/wiki/폐포_(수학)

Sub:
[[폐포연산자,closure_operator]] w rr
}
[[닫힘성,closedness]] =닫힘성,closedness =,closedness 닫힘성 closedness
{
KmsE:closedness ? 없으면
KmsE:closed

표현
Ggl:"closed under"
Natural numbers are closed under addition and multiplication but not subtraction.
[[자연수,natural_number]]s들은, '''closed''' under [[덧셈,addition]] and [[곱셈,multiplication]] but not [[뺄셈,subtraction]].

}
아님 [[닫힘성,closure]]가 적당한 pagename인지?

Ggl:"proofwiki closure"
Ggl:"proofwiki closedness"

MathWorld:Closed
= https://mathworld.wolfram.com/Closed.html

TODO 추가/삭제(정리)

= 정리 =
[[Libre:폐포_(수학)]] = https://librewiki.net/wiki/폐포_%28수학%29

[[Date(2023-09-08T21:15:05)]] 그냥 mw에서 closure 검색해서 보이는 항목들임 (del ok) // TMI: [[Micorosft_Edge]]의 컬렉션 기능 처음써봄
{
Set Closure -- from Wolfram MathWorld
https://mathworld.wolfram.com/SetClosure.html

Existential Closure -- from Wolfram MathWorld
https://mathworld.wolfram.com/ExistentialClosure.html

Tight Closure -- from Wolfram MathWorld
https://mathworld.wolfram.com/TightClosure.html

Transitive Closure -- from Wolfram MathWorld
https://mathworld.wolfram.com/TransitiveClosure.html

Reflexive Closure -- from Wolfram MathWorld
https://mathworld.wolfram.com/ReflexiveClosure.html

... 검색: https://mathworld.wolfram.com/search/?query=closure
}

= tmp bmks ko =
2!=2 :: 해석학, 그 열 번째 이야기 | 거리공간에서의 Interior와 Closure ( Interior and Closure for Metric Space )
https://chocobear.tistory.com/172