Mathematics

Questions tagged [neighbourhood]

A neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.

A neighbourhood topology on a set $X$ assigns to each element $x \in X$ a non empty set $\mathcal N(x)$ of subsets of $X $, called neighbourhoods of $x$, with the properties:

  1. If $N$ is a neighbourhood of $x$ then $x \in N$.

  2. If M is a neighbourhood of $x$ and $M \subseteq N \subseteq X$, then $N$ is a neighbourhood of $x$.

  3. The intersection of two neighbourhoods of $x$ is a neighbourhood of $x$.

  4. If $N$ is a neighbourhood of $x$, then $N$ contains a neighbourhood $M$ of $x$ such that $N$ is a neighbourhood of each point of $M$.

56 questions
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Two microneighbourhoods coincide

In Bell's A primer of infinitesimal analysis , three microneighbourhoods of zero are defined: $M_1 = \{ \epsilon \ |\ \epsilon^2 = 0 \}.$ $M_2 = \{ \epsilon \ |\ \lnot \epsilon \ne 0 \}.$ $M_3 = [0, 0].$ It is immediate that $M_1 \subseteq M_2…
Keplerto
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Should the topology on a quasi-seminormed abelian group be generated by open balls as subbasis or as neighborhood basis?

Let $M$ be an abelian group. A quasi-seminorm on $M$ is a function $|\cdot|:M\to \mathbb{R}_{\geq 0}$ s.t. $|0|=0$ ; symmetry: $|-f|=|f|$ ; generalized triangle inequality: $\exists K\geq1,\forall f,g\in M,|f+g|\leq K(|f|+|g|)$ . It is a seminorm if…
Z Wu
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What's the expected maximum indegree of a nearest neighbor graph of $n$ nodes?

There are $3^{14} = 4782969$ cities on the magical planet π. Each city has a teleportation station, and the distance between any two stations is not equal. If all the cities send diplomatic ambassadors from this station to the nearest station at the…
138 Aspen
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Homeomorphism between neighborhoods in topological groups

Let $G$ and $H$ be topological groups and $U$ , $V$ neighborhoods of $e$ in $G$ and $H$ respectively. Note we are not assuming neighborhoods are open. Suppose $p:G\to H$ is a continuous group homomorphism such that $p|_U:U\to V$ is a homeomorphism.…
Ook
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Does a stretch of a set $E\subset\mathbb{R}$ about a condensation point of $E$ intersect $E?$

A point $p$ in a metric space $X$ is said to be a condensation point of a set $E\subset X$ if every neighbourhood of $p$ contains uncountably many points of $E.$ [Note that $p$ need not be in $E.$ ] Suppose $0$ is a condensation point of a set…
Adam Rubinson
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Why basis of topological space is union of local basis of topological space?

Let $(X,O)$ be topological space which is defined by local basis(fundamental neighborhood system) $B_x$ at $x∈X$ . Then, why $∪_{x∈X}B_x$ is basis of $X$ , in other words, every open set of $(X,O)$ is written by union of $∪_{x∈X}B_x$ ? I think this…
Poitou-Tate
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Standardized radius of validity of normal coordinates in differential geometry?

Depending on the curvature of a certain manifold $M$ , intuitively it is interesting to quantify how quickly normal coordinates defined at a point $p$ deteriorate away from $p$ . Is there a standardized measure to quantify the actual size of the…
Kagaratsch
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Minkowski sum of the set of irrational with an open interval is open?

"If $A$ and $B$ are subsets of the set of real numbers where either $A$ or $B$ is open, then the Minkowski sum of $A$ and $B$ is open". I am failing to see how it can be true, as any real number belonging to an open interval, say $(0,1)$ when added…
Rotnap Marsha
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Clarification in definition of neighborhood in Topology

Suppose a topology $\tau$ is given on a set $X=\{a,b,c\}$ . Let $\tau=\{\emptyset, X,\{a\}\}$ . Let $a\in X$ . Can I say that $\{a\}$ is a neighborhood of $a$ ? Because of the fact that $a$ lies in $\{a\}$ . Kindly help.
HPS
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Confusion of definition of neighbourhood space Mendelson Chapter $3$

Something is bugging me about Mendelson's Introduction to Topology Chapter $3$ Definition $3.4.$ Quote from the book: "Definition $3.4:$ Let $X$ be a set. For each $x\in X,$ let there be given a collection $\mathfrak{N}_x$ of subsets of $X$ (called…
Adam Rubinson
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Equivalence of isolation and infinite neighborhood in Metric Spaces.

Let $M$ be a metric space and and $x$ a point in $M$ . I have two statements: (1) $x$ not isolated and (2) Every neighborhood of $x$ contains an infinite number of points of M. I want to prove that stament 1 and 2 are equvalent: 1 $\implies$ 2 We…
Mathstudent123
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Use of the last axiom of neighbourhoods' topology proving equivalence with open sets

I'm proving that the standard definition by open sets of a topology (closed under arbitrary unions and finite intersections) is equivalent to the definition by neighbourhoods. I'll give the precise definition I'm using below. This is where I'm at:…
Daniel Tobar
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intersection of an infinite number of set interiors vs interior of intersection of infinite number of sets

I started reading Topology and Groupoids by Ronald Brown . The context for the following is the real line, neighborhoods and interiors are (at this point) defined using open balls around points on the real line using euclidean distance $|a-b|$ . The…
Marcus Junius Brutus
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Equivalence of two different formulations of neighborhood systems in a set

When defining a non-empty neighborhood system $\mathcal V _x$ of a set $X$ , the formulation of some authors differ. Some provide these $4$ criteria: $\forall U \in \mathcal V _x , x\in U$ $\forall U,V \in \mathcal V _x , U\cap V \in \mathcal V _ x$…
niobium
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Closed subgroups of profinite groups and basis of neighbourhoods

Let $G$ be a profinite group with a basis of neighbourhoods $U_n$ of normal subgroups. Furthermore let $H\subset G$ be a closed subgroup. Then we can define the open subgroup $ H_n:= H\cdot U_n$ . Is it true that $\cap H_n = H$ ? Clearly $H$ is…
Notone
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