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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 $M$.

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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…
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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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 _…
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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Question about the “thermodynamic order‐topology”

This question is inspired by thermodynamics. To give the setting, we first must understand the basic principle behind the second law. Suppose you have two systems: system $A$ at temperature $T_1$ and system $B$ at temperature $T_2$. Denote the flow…
Markus Klyver
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Neighbourhood basis for topology of uniform space.

I've recently encountered the concept of uniform space in topology. I'm not familiar with it at all and I am trying to understand things a bit better. Let $U$ be a uniform structure on $X$. For $x\in X$ and $W\in U$, let $W(x) = \{y:(x,y)\in W\}$.…
ECL
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