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.

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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,...
IgotYourPoint's user avatar
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1 answer
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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^{-...
BBadman's user avatar
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3 votes
1 answer
88 views

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 ...
murcho's user avatar
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3 votes
3 answers
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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{...
achui's user avatar
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1 answer
45 views

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 $q\in X$...
Turquoise Tilt's user avatar
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1 answer
23 views

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 ...
General Mathematics's user avatar
2 votes
1 answer
79 views

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 ...
General Mathematics's user avatar
0 votes
5 answers
73 views

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 $X$ ...
Irving Rabin's user avatar
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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, ...
Jakobian's user avatar
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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 ...
Christopher Dearlove's user avatar
2 votes
1 answer
61 views

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)|...
ferolimen's user avatar
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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\}, $...
Saaqib Mahmood's user avatar
11 votes
1 answer
125 views

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 ...
Jakobian's user avatar
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8 votes
1 answer
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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 $...
caffeinemachine's user avatar
2 votes
2 answers
477 views

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 ...
Emo's user avatar
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3 votes
1 answer
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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) = \...
Pedro Lourenço's user avatar
4 votes
1 answer
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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 \...
George Coote's user avatar
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2 votes
3 answers
147 views

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, $\{x\}\cup\bigcup_{n<\omega}(\omega\setminus f(n))\times\{n\}$ ...
Steven Clontz's user avatar
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1 answer
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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 ...
Chingis's user avatar
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3 answers
107 views

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 ...
Steven Clontz's user avatar
1 vote
1 answer
34 views

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 ...
Vinay Deshpande's user avatar
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3 answers
42 views

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 ...
Pedro Lourenço's user avatar
4 votes
1 answer
115 views

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 ...
D.R.'s user avatar
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3 votes
1 answer
74 views

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 ...
porridgemathematics's user avatar
2 votes
2 answers
94 views

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 \...
Praveen Kumaran P's user avatar
5 votes
1 answer
89 views

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 $b$...
Rio Joy's user avatar
  • 53
2 votes
2 answers
108 views

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 ...
Steven Clontz's user avatar
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1 answer
44 views

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 ...
user avatar
2 votes
0 answers
40 views

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 $\...
obitobi_tobias's user avatar
3 votes
0 answers
63 views

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: ...
Runyang Wang's user avatar
2 votes
1 answer
36 views

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 ...
nkh99's user avatar
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0 votes
0 answers
49 views

$\Bbb R\setminus \Bbb Q$ 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
Arghya Santra's user avatar
0 votes
2 answers
38 views

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 ...
Kevinlove's user avatar
2 votes
1 answer
82 views

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) = ...
porridgemathematics's user avatar
1 vote
0 answers
40 views

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 ...
Jerry Guern's user avatar
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0 answers
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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 ...
B.Casals's user avatar
1 vote
0 answers
30 views

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 ...
Jhon C's user avatar
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-3 votes
1 answer
67 views

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,...
Arghya Santra's user avatar
1 vote
1 answer
72 views

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 ...
psychicmachinist's user avatar
3 votes
1 answer
92 views

$A \subset \mathbb R$ 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 ...
sunny's user avatar
  • 649
2 votes
3 answers
282 views

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 ...
Toilet Paper's user avatar
0 votes
1 answer
33 views

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 ...
Remco van der Zwaag's user avatar
2 votes
1 answer
41 views

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 ...
Ryukendo Dey's user avatar
0 votes
2 answers
70 views

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 $f, g: X \to Y$ ...
Ryukendo Dey's user avatar
0 votes
1 answer
34 views

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-...
Cartesian Bear's user avatar
3 votes
1 answer
144 views

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 ...
Asigan's user avatar
  • 697
1 vote
1 answer
62 views

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 $x$....
Ryukendo Dey's user avatar
2 votes
0 answers
68 views

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/...
P.Jo's user avatar
  • 714
3 votes
1 answer
138 views

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 ...
zyy's user avatar
  • 967
5 votes
1 answer
66 views

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 $n^2$,...
Jingjin Wang's user avatar

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