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.
33
questions
1
vote
1
answer
51
views
Convergence of Sequences with respect to Ultrafilters on the Natural numbers in Compact Hausdorff Topological Spaces
Let $(X,\mathcal T)$ be a topological space.
Let $\mathcal F$ be an ultrafilter on $\mathbb N$.
We say that sequence $(x_n)_{n\in\mathbb N}$ converges with respect to $\mathcal F$ to $x$, iff for all ...
- 53
0
votes
1
answer
36
views
Reconciling metric and topological neighborhoods
Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological ...
- 119
0
votes
0
answers
16
views
Neighbouring cover in a directed graph
I am not sure if this type of problem has been studied before, so it would be great to receive some guidance. Consider a directed graph G=(V,E). We define an in-neighbourhood cover as a subset $W\...
1
vote
0
answers
49
views
Extension topology
I am reading a paper by Goldman and Sah on extension topology, but I am uncertain about the meaning of the sentences. Paraphrasing the paper:
Let $X$ be an abelian group and $M$ be a subgroup. ...
- 151
0
votes
0
answers
21
views
Set notation meaning of $U_{1/n} (\xi)$
I am studying real analysis and do not understand the notation:
The set $U_{1/n}(\xi).$ Is this the same as $(\xi-1/n; \xi+1/n)$? Thank you for your help.
I will attach the textbook passage for more ...
- 1
1
vote
1
answer
44
views
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 ...
- 177
1
vote
1
answer
54
views
How to prove this proposition about collection of continuous functions $\{f_i\}_{i \in I}$?
Let $(X,\tau)$ be a topological space. Suppose that $\{f_i\}_{i \in I}$ is a collection of continuous functions $X \rightarrow \mathbb{R}$ such that for every $x \in X$, there exists a neighborhood $...
- 476
1
vote
1
answer
71
views
Understanding the axioms for neighbourhoods and their independence/consistency
After reading answers to this question I can't help being confused about axioms 2 and 4 being truly independent, or about their true meaning for that matter.
Recalling them, for a given set $X$:
If $...
0
votes
0
answers
23
views
Prove that a vertex with minimum degree necessarily a vertex cut
From Graphs and Digraphs (7 ed.), proof of Theorem 4.6:
If $G$ is not complete, then take a vertex $v$ with $\deg{v} = \delta(G)$ and note that $N(v)$ is a vertex-cut of $G$.
This is stated without ...
- 927
2
votes
2
answers
141
views
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 ...
1
vote
0
answers
46
views
Is this condition redundant (neighborhood filter for TVS in Trèves)?
In Trèves's Topological Vector Spaces, Distributions and Kernels, Theorem 3.1 is as follows.
A filter $\mathscr F$ on a vector space $E$ is the filter of neighborhoods of the origin in a topology ...
- 6,734
0
votes
0
answers
140
views
Linear topology
Definition: A linear topology $\tau$ on a left $A$-module $M$ is a topology on $M$ that is invariant under translations and admits a fundamental system of neighborhood of $0$ that consists of ...
- 151
0
votes
1
answer
52
views
Identifying Non-Analytic Regions for the Function $f(z) = \frac{1}{{z^2 + 5iz - 4}}$
I'm working with the complex function $f(z) = \frac{1}{{z^2 + 5iz - 4}}$, and I'm trying to determine where this function is not analytic. I've been trying to compute its domain of analyticity, but I'...
- 1
0
votes
1
answer
187
views
Let A and B be subsets of a topological space (X, τ). Prove that $\overline{A\cap B} \subseteq \overline{A} \cap \overline{B}$
Hi I was working on the following problem :
I have some kind of intuition why this would be (tough I might be wrong) :
We can break up the $ \overline{A \cap B} $ into $ A \cap B$ and all L the set ...
- 348
-1
votes
1
answer
34
views
Count neighbors at radius $r_m$
I have a point cloud of $50,000$ points in a 3D space.
I want to count the number of neighbors each point $\{(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3), ... ...,(x_N,y_N,z_N)\}$ has at radii $\{r_1, ...
- 1,671