Neighborhood

On a plane, a subset

V

{\ displaystyle V}

V

is a neighborhood of the point

p

{\ displaystyle p}

p

if around the point you can draw a small disk that will be entirely contained in

V

{\ displaystyle V}

V

.

A rectangle cannot be a neighborhood of its vertices.

A neighborhood of a point is a set containing a given point and close (in some sense) to it. In different sections of mathematics, this concept is defined in different ways.

Content

1 Definitions
- 1.1 Mathematical analysis
- 1.2 General Topology
2 notes
3 Example
4 Variations and generalizations
- 4.1 Punctured neighborhood
5 See also
6 notes
7 Literature

Definitions

Mathematical Analysis

Let be $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ arbitrary fixed number.

Neighborhood of the point $x_{0}$ ${\ displaystyle x_ {0}}$ on the number line (sometimes they say $\varepsilon$ ${\ displaystyle \ varepsilon}$ -neighborhood) is the set of points remote from $x_{0}$ ${\ displaystyle x_ {0}}$ less than $\varepsilon$ ${\ displaystyle \ varepsilon}$ , i.e $O_{\varepsilon }(x_{0})=\{x:|x-x_{0}|<\varepsilon \}$ ${\ displaystyle O _ {\ varepsilon} (x_ {0}) = \ {x: | x-x_ {0} | <\ varepsilon \}}$ .

In the multidimensional case, the neighborhood function is performed by the open $\varepsilon$ ${\ displaystyle \ varepsilon}$ ball centered at a point $x_{0}$ ${\ displaystyle x_ {0}}$ .

In a Banach space $(B,\|\cdot \|)$ ${\ displaystyle (B, \ | \ cdot \ |)}$ neighborhood centered at $x_{0}$ ${\ displaystyle x_ {0}}$ called a lot $A=\{x\in B:\|x-x_{0}\|<\varepsilon \}$ ${\ displaystyle A = \ {x \ in B: \ | x-x_ {0} \ | <\ varepsilon \}}$ .

In metric space $(M,\rho )$ ${\ displaystyle (M, \ rho)}$ neighborhood centered at $y$ ${\ displaystyle y}$ called a lot $A=\{x\in M:\rho (x,y)<\varepsilon \}$ ${\ displaystyle A = \ {x \ in M: \ rho (x, y) <\ varepsilon \}}$ .

General Topology

Let a topological space be given $(X,{\mathcal {T}})$ ${\ displaystyle (X, {\ mathcal {T}})}$ where $X$ ${\ displaystyle X}$ Is an arbitrary set , and ${\mathcal {T}}$ ${\ displaystyle {\ mathcal {T}}}$ - defined on $X$ ${\ displaystyle X}$ topology .

A bunch of $V\subset X$ ${\ displaystyle V \ subset X}$ called a neighborhood of a point $x\in X$ ${\ displaystyle x \ in X}$ if there is an open set $U\in {\mathcal {T}}$ ${\ displaystyle U \ in {\ mathcal {T}}}$ such that $x\in U\subset V$ ${\ displaystyle x \ in U \ subset V}$ .
Similarly, the neighborhood of the set $M\subset X$ ${\ displaystyle M \ subset X}$ called so many $V\subset X$ ${\ displaystyle V \ subset X}$ that there is an open set $U\in {\mathcal {T}}$ ${\ displaystyle U \ in {\ mathcal {T}}}$ for which $M\subset U\subset V$ ${\ displaystyle M \ subset U \ subset V}$ .

Remarks

The above definitions do not require the neighborhood $V$ ${\ displaystyle V}$ was an open set, but only that it contained an open set $U$ ${\ displaystyle U}$ . Some authors insist that any neighborhood is open. ^[1] Then a neighborhood of a set is any open set containing it. This difference is not fundamental for the development of further topological theory. However, in each case, it is important to fix the terminology.

The neighborhood of many points $M$ ${\ displaystyle M}$ called so many $V$ ${\ displaystyle V}$ , what $V$ ${\ displaystyle V}$ there is a neighborhood of any point $x\in M$ ${\ displaystyle x \ in M}$ .

Example

Let a real line with a standard topology be given. Then $(-1,2)$ ${\ displaystyle (-1,2)}$ is an open neighborhood, and $[-1,2]$ ${\ displaystyle [-1,2]}$ - closed neighborhood of a point $0$ ${\ displaystyle 0}$ .

Variations and generalizations

Pierced Surroundings

A punctured neighborhood of a point is a neighborhood of a point from which this point is excluded.

Strictly speaking, a punctured neighborhood is not a neighborhood of a point, since according to the definition of a neighborhood, a neighborhood must include the point itself.

Formal Definition: Many ${\dot {V}}$ ${\ displaystyle {\ dot {V}}}$ called a punctured neighborhood (punctured neighborhood) of a point $x\in X$ ${\ displaystyle x \ in X}$ , if

{\dot {V}}=V\setminus \{x\},

{\ displaystyle {\ dot {V}} = V \ setminus \ {x \},}

Where $V$ ${\ displaystyle V}$ - neighborhood $x$ ${\ displaystyle x}$ .

Notes

↑ Rudin, 1975 , p. 13.

Literature

Mathematical Encyclopedia. - M .: Soviet Encyclopedia , 1984 . - T. 4.
U. Rudin. Functional analysis. - M .: Mir , 1975 .

[_4771e3f98bbcb7bc-1] Rudin, 1975 , p. 13.