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Neighbourhood (mathematics)

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From DawoumWiki, the free Mathematics self-learning
(Redirected from Neighborhood (topology))
Original article: w:Neighbourhood (mathematics)
For the concept in graph theory, see Neighbourhood (graph theory).
A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} in the plane is a neighbourhood of a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} if a small disc around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V.}

토폴로지(topology)와 관련된 수학(mathematics)의 영역에서, 이웃(neighbourhood 또는 neighborhood)은 토폴로지적 공간(topological space)에서 기본 개념 중 하나입니다. 그것은 열린 집합(open set)과 내부(interior)의 개념과 밀접한 관련이 있습니다. 직관적으로 말하자면, 점의 이웃은 그 점을 포함하는 점들의 집합(set)으로, 집합을 떠나지 않고 그 점에서 멀어지는 방향으로 어느 정도 이동할 수 있습니다.

Definitions

Neighbourhood of a point

만약 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 가 토폴로지적 공간(topological space)이고 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 에서 한 점이면, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 의 이웃은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 를 포함하는 열린 집합(open set) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} 를 포함하는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 의 부분집합(subset) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 입니다: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p \in U \subseteq V \subseteq X.}

이것은 역시 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 에서 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 의 토폴로지적 내부(topological interior)에 속하는 점 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p \in X} 와 동등합니다.

이웃 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 는 열린 부분집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 일 필요는 없지만, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 에서 열린 것일 때 그것은 열린 이웃이라고 불립니다.[1] 일부 저자는 이웃을 열린 것으로 요구하는 것으로 알려져 있으므로, 규칙을 기록하는 것이 중요합니다.

각 점의 이웃인 집합은 각 점을 포함하는 열린 집합의 합집합으로 표현될 수 있기 때문에 열린 것입니다. 그림에 표시된 것처럼, 직사각형은 모든 점의 이웃이 아닙니다; 사각형의 가장자리 또는 모서리에 있는 점은 직사각형 내에 포함된 임의의 열린 집합에 포함되지 않습니다.

A closed rectangle does not have a neighbourhood on any of its corners or its boundary.

점의 모든 이웃의 모음은 그 점에서 이웃 시스템(neighbourhood system)이라고 불립니다.

Neighbourhood of a set

만약 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 가 토폴로지적 공간 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 의 부분집합(subset)이면, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 이웃은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 를 포함하는 열린 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} 를 포함하는 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 입니다,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S \subseteq U \subseteq V \subseteq X.} 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 이웃인 것과 그것이 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 에서 모든 점의 이웃인 것은 필요충분 조건임을 따릅니다. 게다가, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 이웃인 것과 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 의 내부(interior)의 부분집합인 것은 필요충분 조건입니다.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 의 열린 부분집합이기도 한 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 이웃은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 열린 이웃이라고 불립니다. 점의 이웃은 단지 이 정의의 특별한 경우입니다.

In a metric space

A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} in the plane and a uniform neighbourhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S.}
The epsilon neighbourhood of a number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle a} on the real number line.

메트릭 공간(metric space) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle M = (X, d)} 에서, 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 는 다음이 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 에 포함됨을 만족하는 중심 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 와 반지름 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r>0} 을 갖는 열린 공(open ball)이 존재하면 점 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 의 이웃입니다: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle B_r(p) = B(p; r) = \{ x \in X : d(x, p) < r \}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 모든 원소 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 에 대해, 다음이 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 에 포함됨을 만족하는 양수 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} 이 존재하면 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 균등 이웃이라고 불립니다: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle B_r(p) = \{ x \in X : d(x, p) < r \}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r > 0} 에 대해, 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 의 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} -이웃 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S_r} 은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 에서 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} 보다 작은 거리에 있는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 에서 모든 점의 집합입니다 (또는 동등하게, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S_r} 은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S} 에서 한 점에 중심을 둔 반지름 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} 의 모든 열린 공의 합집합입니다): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S_r = \bigcup\limits_{p\in{}S} B_r(p).} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} -이웃은 균등 이웃이고, 집합이 균등 이웃인 것과 그것이 일부 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} 값에 대해 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle r} -이웃을 포함하는 것은 필요충분 조건임을 직접 따릅니다.

Examples

The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M.

보통의 유클리드 메트릭(Euclidean metric)과 다음으로 정의된 부분집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 를 갖는 실수 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle \R} 의 집합이 주어지면, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V := \bigcup_{n \in \N} B\left(n\,;\,1/n \right),} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 는 자연수의 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle \N} 에 대해 이웃이지만, 이 집합의 균등 이웃은 아닙니다.

Topology from neighbourhoods

See also: Filters in topology, Topological space § Neighborhood definition, and Axiomatic foundations of topological spaces § Definition via neighbourhoods

위의 정의는 열린 집합(open set)의 개념이 이미 정의되어 있으면 유용합니다. 토폴로지를 정의하는 대안적인 방법이 있으며, 먼저 이웃 시스템(neighbourhood system)을 정의하고, 그런-다음 각 점의 이웃을 포함하는 집합으로 집합을 정의하는 것입니다.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 위에 이웃 시스템은 다음을 만족하는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 에서 각 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle x} 에 대한 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 의 부분집합 필터(filter) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle N(x)} 의 할당입니다:

  1. 점 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle x} 는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle N(x)} 에서 각 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} 의 원소입니다.
  2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle N(x)} 에서 각 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} 는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 에서 각 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle y} 에 대해, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle N(y)} 안에 있음을 만족하는 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle N(x)} 에서 일부 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 를 포함합니다.

우리는 두 정의가 호환 가능하다는 것을 보여줄 수 있습니다. 즉, 열린 집합을 사용하여 정의된 이웃 시스템에서 얻은 토폴로지가 원래 것이고, 이웃 시스템에서 시작할 때 그 반대도 마찬가지입니다.

Uniform neighbourhoods

균등 공간(uniform space) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle S = (X, \Phi)} 에서, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 는 만약 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle V} 가 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle P} 의 일부 점에 대한 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U} -닫힌 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle X} 의 모든 점을 포함함을 만족하는; 즉, 모든 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle x \in P} 에 대해 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U[x] \subseteq V} 임을 만족하는 측근(entourage) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle U \in \Phi} 가 존재하면 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle P} 의 균등 이웃이라고 불립니다.

Deleted neighbourhood

점 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 의 삭제된 이웃 (때때로 구멍난 이웃(punctured neighbourhood)이라고 불림)은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle \{p\}} 없이 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p} 의 이웃입니다. 예를 들어, 구간(interval) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle (-1, 1) = \{y : -1 < y < 1\}} 은 실수 직선에서 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle p = 0} 의 이웃이므로, 집합 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle (-1, 0) \cup (0, 1) = (-1, 1) \setminus \{0\}} 은 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle 0} 의 삭제된 이웃입니다. 주어진 점의 삭제된 이웃은 사실 그 점의 이웃이 아닙니다. 삭제된 이웃의 개념은 함수의 극한의 정의와 (다른 것 중에서) 극한점 정의에서 발생합니다.[2]

See also

References

  1. ^ Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Sterling K. Berberian. Springer. p. 6. ISBN 0-387-90972-9. According to this definition, an open neighborhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle x} is nothing more than an open subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle E} that contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:7231/localhost/v1/":): {\displaystyle x.}
  2. ^ Peters, Charles (2022). "Professor Charles Peters" (PDF). University of Houston Math. Retrieved 3 April 2022.{{cite web}}: CS1 maint: url-status (link)

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