Questions tagged [connectedness]
Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.
3,019
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Connectedness of manifolds lying in a cylinder with a hole
Problem setting: We define a cylinder with a hole
\begin{equation}
A_{\delta} = \{(x',x_N) \in \mathbb{R}^{N-1} \times \mathbb{R} \mid \delta \leq |x'| \leq 1 \}
\end{equation}
for some $\delta \in (0,...
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Baby Rudin Exercise 2.21
The set up is as follows:
"
Let $A$ and $B$ be separated subsets of some $\mathbb{R}^k$, and suppose $a \in A$, $b \in B$, and define
$p(t) = (1-t)a + tb$, for $t \in \mathbb{R}$. Put $A_0 = p^{-...
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Connectedness of the boundary of a domain
I've been struggling to prove the following lemma: "Let $\Omega\subset\mathbb{R}^{d}$ be open and bounded with a Lipschitz boundary and such that $\mathbb{R}^{d}\setminus\partial\Omega$ has ...
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set connectedness with two equivalent definition. how to prove they are equal?
I've seen two different and perhaps equivalent definitions of connected sets.
set $E$ is connected when.
$\nexists{C, D}$ such that both open and $C \cap D = \emptyset$ & $E = C \cup D$
$\nexists{...
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Connected components problem
Let $X$ a set and $p \in X$ a point, and define $\tau$ a topology as
$$\tau := \{ A\subset X: p \in X-A \;\text{or}\; X-A\;\text{is finite}\}$$
Find connected components $C(q)$ for every point $q\in X$...
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Is $A$ is disconnected if some subset of $A$ has separation?
I have asked a question here, Show that, Union of the point (0, 0) with closed line segments between $(0, 0)$ & $(1, 1/n)$ for all $n\in\mathbb{N}$ is connected
In the second part of answer,that ...
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Show that, Union of the point (0, 0) with closed line segments between $(0, 0)$ & $(1, 1/n)$ for all $n\in\mathbb{N}$ is connected
Question: Show that, Union of the point (0, 0) with closed line segments between $(0, 0)$ & $(1, \frac{1}{n})$ for all $n\in\mathbb{N}$ is connected.
My attempt: I want to show this with general ...
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Intuitive reason that connected spaces force discrete valued maps to be constant.
Precursor definitions and proposition, from Bredon's Topology and Geometry.
Definition: Let $X$ be a topological space. A separation of $X$ is a pair $U,V$ of disjoint, nonempty, open subsets of $X$ ...
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Does there exist a second-countable locally connected space with no countable basis of connected sets?
Space $X$ is called locally connected if it has a basis consisting of connected sets.
It's called second-countable if it has a countable basis.
If $X$ is both locally connected and second-countable, ...
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Are there any tighter bounds on the mean number of orthogonally connected components in a binary random rectangular grid?
Question 17 of the set of 20 probability problems at https://www.math.ucdavis.edu/~gravner/MAT135A/resources/chpr.pdf provides bounds of $\tfrac{mn}{8}$ and $\tfrac{(m+2)(n+2)}{6}$ for the mean number ...
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Connected component of the constant function $1$ in $C(S^1,\mathbb{C})$
Let $C(S^1,\mathbb{C})$ be the space of continuous functions $f:S^{1} \subset \mathbb{C} \longrightarrow \mathbb{C}$ endowed with the supremum norm, that is, $ \lVert f \rVert= sup_{z \in S^{1}} |f(z)|...
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Is this subspace of $\mathbb{R}^2$ connected? arcwise connected?
Let
$$
A \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, 0 < x \leq 1, \ y = \sin \frac1x \right\}
$$
and
$$
B \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, y=0, \ -1 \leq x \leq 0 \right\},
$...
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Does locally compact, locally connected, connected metrizable space admit a metric with connected balls?
If $X$ is a locally compact, locally connected, connected metrizable space, does that imply that there must be a metric $d$ on $X$ such that $B(x, r) = \{y\in X : d(x, y) < r\}$ is connected for ...
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(Path) connected components of zero-sets of limiting functions.
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\R}{\mathbb R}$
$\newcommand{\bd}{\text{Bd}}$
Let $X$ be a compact metric space and $f_n:X\to \R$ be a continuous function on $X$, one for each $n$.
Assume $...
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Do any two connected spaces have a continuous surjection between them?
It is known that if $X$ is a connected topological space and there exists a continuous surjection $f:X\to Y$, then so is $Y$.
I wonder if there exist connected topological spaces $X$ and $Y$ such that ...
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When is the intersection of all open normal subgroups equal to the connected component at the identity?
I've managed to show that for locally compact abelian groups, the connected component at the identity $G_0$ is equal to the intersection of all open subgroups of G, since we have that $$A(G_0) = \...
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Is there an infinite-dimensional normed vector space in which the set complement of a bounded set may have more than one unbounded component?
Let $n \ge 2$. Using the fact that $\{x \in \mathbf R^n : \lVert x\rVert > R\}$ is connected for each $R > 0$, we can show that if $B \subseteq \mathbf R^n$ is bounded, then $\mathbf R^n \...
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Is the Maximal Compact Topology (Counterexamples #99) extremally disconnected?
The Maximal Compact Topology is the set $\omega^2\cup\{x,y\}$ topologized by the following basis: points of $\omega^2$ are isolated,
$\{x\}\cup\bigcup_{n<\omega}(\omega\setminus f(n))\times\{n\}$ ...
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Prove that product of two locally connected spaces is locally connected. [duplicate]
Let $X$ and $Y$ be two locally connected spaces. I need to show that $Z = X \times Y$ is locally connected. Here is my attempt:
Proof. Let $z=(x, y)\in Z$ and $N$ be any neighborhood of $z$. I need to ...
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What separation is required to ensure extremally disconnected spaces are sequentially discrete?
Conversation continued at MathOverflow
In https://github.com/pi-base/data/issues/387 it is noted that Tychonoff extremally disconnected spaces are sequentially discrete (see Encyclopedia of General ...
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Group of Integer Valued Functions on Interior of Unit Circle
This is from Higson's Analytic K-Homology.
Let $X$ be a non-empty and compact subset of $\mathbb{C}$. An index function for $X$ is an integer-valued function on the set of bounded components of the ...
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Connected subgroups of unit circle
It's well known that any subgroup H of $\mathbb{S}^1$ is either dense or finite. Therefore, if H is compact, it implies that H either is finite or equal to the entire unit circle. My question is ...
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If a closed subset $C \subseteq \mathbb R^2$ is "fragilely"/“minimally” simple connected, must it be all of $\mathbb R^2$?
It is a corollary of the Riemann mapping theorem that every path-connected simply-connected open subset of $\mathbb R^2$ is homeomorphic to $\mathbb R^2$ (there are more elementary proofs as well ...
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Connected components of conformal image of boundary
Let $f : G \rightarrow \mathbb{D}$ be a biholomorphism (a holomorphic map with holomorphic inverse), and suppose $G$ is a bounded open subset of the plane. Let $C$ be a compact connected subset of the ...
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Let A be a countable set of $\mathbb{R}^n , n>1$. We claim that $\mathbb{R}^n-A$ is connected.
The proof goes as follows... "Given $x,y \in \mathbb{R}^n-A,$ Choose a point $z \in [x,y]$ other than $x$ and $y$, where $[x,y]=\{a \in \mathbb{R}^n :a=(1-t)x+ty,0\leq t\leq1\}$, Now choose $w \...
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Strongly connected? Binary numbers of length n with bounded hamming weight
I am wondering if the following is true, and if yes how to prove it:
Construct a graph where the vertices are binary numbers of length $n$ and have a hamming weight (i.e. digit sum) between $a$ and $b$...
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Extremally disconnected without Hausdorff
In Theorem T000045 of pi-Base, a proof is given to defend the assertion from Counterexamples in Topology that all Extremally disconnected ($T_2$ where the closure of open is open) spaces are Totally ...
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Definition of nested partition of the circle
Below is an excerpt from the paper Boundary torsion and convex caps of locally convex surfaces, in which the author defines a so-called nested partition of the circle.
I am having a hard time ...
user242708
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Is ($\mathbb{Q} \times \mathbb{Q}) \cup (\mathbb{R}\setminus \mathbb{Q} \times \mathbb{R}\setminus \mathbb{Q})$ connected [duplicate]
Does anyone know whether or not the set ($\mathbb{Q} \times \mathbb{Q}) \cup (\mathbb{R}\setminus \mathbb{Q} \times \mathbb{R}\setminus \mathbb{Q})$ is connected with the sub space topology from $\...
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0
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Connected Space with uncountably many path components [duplicate]
Here is a question I saw today: Provide an example of a connected topological space having uncountably many path components.
Actually I have came up with an example which I believe should be correct:
...
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The set of coefficients of cubics having three real roots is connected
Let $D$ be the set of all $3-tuples$, $(a,b,c)$ in $R^3$ such that the cubic polynomial $x^3+ax^2+bx+c$ has three real roots, i.e., $D=\{(a,b,c) \in \mathbb{R}^3 \mid x^3+ax^2+bx+c \textit{ factors ...
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$\Bbb R\setminus \Bbb Q$ the set of irrationals is disconnected [duplicate]
I'm trying to find two non-empty disjoint open set $A, B$ such that $A \cup B = \Bbb R \setminus \Bbb Q$.
But can't find
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2
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Let $X=U\cup V$ where $U,V$ open and disjoint, then every connected subset $A$ of $X$ is a subset of $U$ or a subset of $V$.
I am asked to prove that if $X=U\cup V$ where $U,V$ open and disjoint, then every connected subset $A$ of $X$ is a subset of $U$ or a subset of $V$.
My way of solution was assuming in contradiction ...
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Cutting a disk with piecewise analytic curves
Let $B \subset \mathbb{C}$ be an open set and assume that the boundary of $B$ consists of finitely many piecewise analytic curves. Suppose $z_0 \in \partial B$ and assume the following:
$U(z_0 , r) = ...
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How to prove that this polynomial inequality region is connected?
I understand that proof that the Mandelbrot Set is connected is not easy to follow and requires mathematical tools an Engineer probably doesn't have. But I got to wondering if there was a more ...
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Cartesian product of Locally connected spaces is locally connected iff each space is locally connected [duplicate]
I need the proof of the Cartesian product of Locally connected spaces is locally connected iff each space is locally connected. I do have the left implication, as it is trivial using continuity and ...
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Connectedness of $Y \cup A$ and $Y \cup B$
Let $Y$ be a nonempty closed subset of $X \subset \Bbb R^n$. Assume $X\setminus Y = A\cup B$, with $A,B \subset X$ nonempty, open and disjoint. Prove $Y\cup A$ and $Y\cup B$ are connected.
If $B$ is ...
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Is the intersection of two connected set is connected?? [duplicate]
Is the intersection of two connected set is connected??
I think it is connected, I can't find any counter example to prove it disconnected.
My approach is->
A,B connected.
Let A meet B disconnected,...
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vote
1
answer
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Definitions of connectedness on rings
I've encountered a few possible definitions of a "connected ring" and am having some confusion relating them. The first one is defined for any commutative ring:
A commutative connected ring ...
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$A \subset \mathbb R$ is connected $\iff A$ is an interval; Hunter
I am reading John K. Hunter's lecture notes and in chapter 5, there is a theorem about every set in $\mathbb{R}$ being connected if and only if it is an interval. The proof is somewhat intricate, but ...
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If G is a connected Graph with a cycle, if you remove an edge from the cycle then G is still connected
The proof is quite obvious but how do you write it down nicely? I have this problem a lot with graph theory proofs where I don't know how to write them down so that they are rigorous enough.
The proof ...
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Definition of connected sets (Baby Rudin)
In Baby Rudin a subset E of a metric space X is defined to be connected if "E is not a union of two nonempty separated sets."
This may be nit picking, but should this not be
E is not a union ...
2
votes
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answer
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On the size of a certain topology defined by an inclusion function.
I found this problem in chapter 7 of biglist. This is supposed to be an easy problem, but I have no idea how to approach it. The problem goes as follows:
Let $(X, \mathcal{T})$ be a topological space ...
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Let $(X, \mathcal{T})$ be topological space and let $(Y, \mathcal{U})$ be Hausdorff Topological space...
The question goes like this:
Let $(X, \mathcal{T})$ be topological space and let $(Y, \mathcal{U})$ be Hausdorff Topological space. Let $A \subseteq X$ be a non-empty subset. Suppose $f, g: X \to Y$ ...
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Showing that $\{x\in A: f(x)\in(a,b)\},A=\{x\in [0,1]:(-f'(x)/f(x)) > c\},b>a,c>0$ is covered by a connected component of $(-f'/f)^{-1}(c/2,+\infty)$
Let $f:\mathbb{R}\to (0,+\infty)$ be a continuously differentiable function on $(-\epsilon, 1 + \epsilon)$ for some $\epsilon > 0$ and piecewise smooth on the rest of $\mathbb{R}$ which is non-...
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In an order topology, are connected sets convex, and are they intervals?
Problem: $X$ is an ordered set with order topology. Is it true that (1) $A\subseteq X$ is connected $\implies$ $A$ is convex (2) $A\subseteq X$ is connected $\implies$ $A$ is an interval ? (Here ...
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1
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Relationship between continuity and sequential continuity [duplicate]
Let $(X, \mathcal{T})$, $(Y, \mathcal{U})$ be topological spaces and let $f: X \rightarrow Y$.
(a) Suppose $f$ is continuous and $\{x_n\}_{n=1}^{\infty}$ is a sequence in $X$ converging to a point $x$....
- 433
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votes
0
answers
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Simply connected open set excluding a point and containing a given compact set
Given a compact set $K$ in $\mathbb{R}^2$ assume that $0$ lies in the unbounded component of $K^c$ (the complement of $K$).
I am struggling to show the following claim and I would appreciate any help/...
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Munkres' proof that topologist's sine curve is not path connected
I was reading Munkres' proof that topologist's sine curve is not path connected. What confused me was why he bothered to reduce the path $f:[a,c]\to\bar{S}$ to $f:[b,c] \to \bar{S}$. As is understood ...
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How many distinct simply connected shapes can be represented on an n by n binary image?
This is a problem which I found to be quite challenging. I ask this because I want to be able to losslessly compress simply connected shapes (connected shapes without holes) with fewer bits than $n^2$,...
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