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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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Connected Metric Spaces: Strategies

I am not really sure if my ideas in this topic are correct. Can anyone help me? Finding the connected components of a metric space $X$. Suppose there are two connected components $C_1, C_2$ of $X$. ...
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$H:M\rightarrow \mathbb{R}$ is a continuous map from top space $M$. Show that $H^{-1}(e)$ divides $M$ where $e\in \text{int}(H(M))$

Let $M$ be a topological space and $H:M\rightarrow \mathbb{R}$ be continuous and surjective. Suppose $e\in \text{int}(H(M))$. Then show $H^{-1}(e)$ divides $M$; that is, $M\setminus H^{-1}(e)$ has ...
Ali's user avatar
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2 answers
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How does this set equality follow?

This is proposition 4.9 in Lee's intro to topological manifolds. What I don't understand is the proof of part b where it is said that $$ X = \overline{A} \subseteq{\overline{U}} = U $$ In particular, ...
Jeff8770's user avatar
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Non local connected $\implies$ Non-open component example [closed]

I'm trying to figure out various definitions of local connectedness, here, proved equivalence of definitions of locally connected space. A topological space $T=(𝑆,\tau)$ is locally connected if and ...
poker resources's user avatar
4 votes
1 answer
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Showing particular quotient space is (not) compact, connected and Hausdorff

I came across the following question, I'm unsure about some of my answers. Let $U = \{(x, y) \in \mathbb{R}^{2} \mid y \in \{0, 1\}\}$ be a subspace of $\mathbb{R}^{2}$. We define the following ...
JLGL's user avatar
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1 answer
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Are these Cayley graphs strongly connected?

My Background: I have no formal training in graph theory. I'm just a group theory PhD student. The Story: I attended the New Perspectives in Computational Group Theory conference last week in honour ...
Shaun's user avatar
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Differential Forms and Path Integrals

How to prove this problem from Fulton Book (Algebraic Topology) Show that an open set U in the plane is connected if and only if there is a segmented path between any two pints of U. Can you show that ...
Haxhi Dacaj's user avatar
2 votes
2 answers
57 views

Proving that Intervals are connected in $\Bbb R$ using continuous functions into $\{0,1\}$

I have come up with a proof for the fact that Intervals are connected sets in $\Bbb R$, that utilizes an equivalent definition of connected sets. Namely: Definition. A topological space $X$ is called ...
clorx's user avatar
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2 votes
3 answers
174 views

Connect two points on an annulus

I want to prove that the annulus given by the set $$ B(a;R_1,R_2) =\{{z \in \mathbb{C} : R_1 < |z-a| < R_2\}} $$ is a connected space. As I am working in $\mathbb{C}$ it is enough to prove it to ...
baristocrona's user avatar
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Is Rudin's definition of simple connectedness in his RCA equivalent to the usual one?

At the bullet point $10.38$ in his Real and Complex Analysis, Rudin gives a very quick rundown on homotopy and simple connectedness, but the definition he gives is the following: If [a topological ...
Bruno B's user avatar
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4 votes
1 answer
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Is a connected space possible if it is Hausdorff? (T2)

Getting into topology this semester and have been studying and doing some exercises and I was wondering if a Hausdorff space i.e: $ {\displaystyle \forall \,x,y\in X\,|\,x\neq y\,\exists \,U,V{\text{ ...
Jorge Ávila Balmaceda's user avatar
1 vote
1 answer
38 views

Continuous map to a connected set

I am trying to solve this problem. I am studying for an exam. Let $\Omega \subseteq \mathbb{R}^n$ be a connected open set. Given $x_0, y_0 \in \Omega$, show that for every $M > 0$ there exists a ...
user123456's user avatar
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1 answer
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Intersection of decreasing sequence of star-convex sets

A set $A \subseteq \mathbb{R}^{n}$ is star-convex if there is $z\in A$ such that for all $x ∈ A$, the line segment from $z$ to $x$ lies in $A$. Let $\{A_i : i \in \mathbb{N}\}$ be a decreasing ...
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Is this an equivalence of connectedness in $\mathbb{R}^{n}$?

Let $A$ be a subset of $\mathbb{R}^{n}$ with the following property: for every $B\subseteq \mathbb{R}^{n}$, $A\cap Bd(B)\neq \emptyset$ (i.e., $A$ has non-empty intersection with the boundary of $B$) ...
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How are the hyperreals totally disconnected

I understand that the rationals are totally disconnected by the fact, that they can form a pair of open sets with an irrational bound whose union is all of $\mathbb Q$. I also see that the hyperreals ...
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How many connected components does the space of continuous functions $C[S_n, S_m]$ have?

For each $n \geq 0$ let $I_n$ denote the closed interval $[2n,2n+1]$ in $\mathbb R$, and equip $$ S_n = \bigcup_{k = 0}^{n} I_k$$ with the subspace topology induced by the standard one on $\mathbb{R}$....
RDL's user avatar
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1 answer
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Is this an equivalence of connectedness?

Let $X$ be a Hausdorff topological space and $A\subseteq X$. Suppose that for every $B\subseteq X$, $A\cap Bd(B)\neq \emptyset$ (i.e., $A$ has non-empty intersection with the boundary of $B$) whenever ...
Peluso's user avatar
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1 vote
1 answer
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Construct a foliation on a simply connected compact manifold with all leaves non-compact [closed]

Here's my progress so far in this problem: There are no codimension-one foliations on even dimensional spheres (here's why). Also, by Novikov's theorem any codimension-one foliation on a simply ...
danimalabares's user avatar
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Let $G$ be an $r$-regular graph with $|V(G)| = 2r + 1$. Prove that $\lambda(G) = r$ (edge-connectivity).

Let $G$ be an $r$-regular graph with $|V(G)| = 2r + 1$. Prove that $\lambda(G) = r$ (edge-connectivity). On the exam, I solved the problem in one direction by considering the fact that in any graph $\...
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Is there a classification of connected sets in $\mathbb{R}^2$? [duplicate]

We know that in $\mathbb{R}$ with standard topology, a subset $A$ of $\mathbb{R}$ is connected if and only if $A$ is an interval (including points, rays, $\mathbb{R}$). Is there a classification of ...
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Numner of Graph functions

Let, T(V, E) be an undirected tree with n vertices, where V={$v_1,v_2,...,v_n$}. Let d(i)represent the degree of verticex $v_i$. Let f be a function from V to {1,2,...,n}. Compute the number of such ...
Arnab Seal's user avatar
3 votes
2 answers
103 views

There exist a path of length $\chi(G)- 1$ in a connected graph G

For any undirected connected graph $G$, let $\chi (G) $ be its chromatic number. Then for every vertex $v$ in $G$, there exists a path of length $\chi (G) - 1$ starting from $v$. My approach : for ...
Arnab Seal's user avatar
2 votes
3 answers
81 views

A countable metric space implies totally disconnectedness

I'm stuck on a problem in general topology. But first, let's establish some observations: Let $(X, \tau)$ be a topological space. The connected component of $x \in X$, denoted by $C(x)$, is the union ...
Joel Marques's user avatar
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46 views

Given the sets $X:(x+1)^2 + y^2 \leq 4, x^2+y^2 \ge 1$ and $Y:1 \leq x^2 + y^2 \leq 4$. Is X homeomorphic to Y?

I'm trying to prove they aren't via connectedness. I figured the point (1,0) is an important point (cut point) in this problem, but i think that X without (1,0) is still connected.
Alejandro Hernando's user avatar
3 votes
0 answers
117 views

Is $\mathbb R/\mathbb N$ homeomorphic to $\mathbb R/\mathbb Z?$

Where $\mathbb R/\mathbb N$ is the quotient topological space, where $\mathbb N$ is collapsed onto a point. Same with $\mathbb R/\mathbb Z.$ I was thinking I could prove it wasn't homeomorphic using ...
Alejandro Hernando's user avatar
6 votes
2 answers
225 views

Can the Euclidean unit interval have a finer connected topology?

Consider the unit interval $X = [0,1]$ equipped with the Euclidean topology $\tau_E$, and consider some other topology $\tau_F$ on $X$ that is strictly finer than $\tau_E$. Does there exist such a ...
aghostinthefigures's user avatar
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12 views

What can't be added for "maximally connected graph" to keep some property said in definition?

In wikipedia it says A graph is said to be maximally connected if its connectivity equals its minimum degree. A graph is said to be maximally edge-connected if its edge-connectivity equals its ...
An5Drama's user avatar
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3 votes
1 answer
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Accessibility Components does *not* coincide with Path Connected and Connected Components for Length Spaces

I am working through "A Course in Metric Geometry" and I do not believe this exercise to be true: Exercise $\boldsymbol{2.1.3}$ $3)$ Verify that accessibility components coincide with both ...
2oovy's user avatar
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6 votes
1 answer
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Is there a nondiscrete linearly ordered topological space (LOTS) that is extremally disconnected?

A space is extremally disconnected if the closure of every open set is open. I was wondering if there is a ordered space that is extremally disconnected but not discrete. Currently there is no such ...
Jianing Song's user avatar
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0 votes
3 answers
67 views

Connected subsets of $\mathbb{Q}^{n}$ have at most one point

Given a set $A \subset \mathbb{Q}^{n}$ with #$A > 1$, is it possible to find $s \in \mathbb{R}^n$, $r \in \mathbb{R}_{> 0}$ such that $a_1 \in B(s,r), a_2 \in \operatorname{int}(\mathbb{R}^{n} - ...
huh's user avatar
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0 votes
1 answer
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Property of continuous function over a compact and connected set

If $A\subset\mathbb{R}^n$ is non-empty, compact and connected. If $f\in C(A,\mathbb{R})$ and $m\in \mathbb{N} $, choose arbitrary points $x_1,...,x_m\in A$. Show that there exists $\alpha\in A$ such ...
user926356's user avatar
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1 vote
0 answers
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Non trivial open sets of a connected hausdorff space are not compact.

I realize that open sets in $\mathbb{R}^n$ under standard topology are not compact. I was then wondering under what conditions can a topology have this be true in general. I then hypothesize that ...
K. D.'s user avatar
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1 vote
2 answers
64 views

Are singletons in $\mathbb{R} \setminus \mathbb{Q}$ both closed and open?

For context: I am looking at $\mathbb{R} \setminus \mathbb{Q}$ as a subset of $\mathbb{R}$ with usual topology. I know the singletons are closed. But are they open? I thought about something along the ...
metamathics's user avatar
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2 answers
41 views

definition of connected space

A space $X$ is connected if it cannot be written as the union of two or more disjoint open sets. Implicitly here there is a finiteness condition, right? As in, if I write down a partition of $X$ with ...
node196884's user avatar
5 votes
2 answers
162 views

A locally compact, connected, Hausdorff and locally connected space is not the countable disjoint union of nonempty closed subsets

In Chapter $5$, Section $4$ of the book Continuum theory by Sam B. Nadler, Jr., the auther defines $\sigma$-connectedness (Definition $5.15$) to be the property of not being a disjoint union of ...
Jianing Song's user avatar
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2 votes
1 answer
338 views

Can a connected planar graph have 10 vertices and edges? is this possible?

Can a connected planar graph have 10 vertices and edges? is this possible? Using Euler’s formula, $V − E + F = 2$. $10 − 10 + F = 2$, Therefore $F = 2$. Do I also need to use this formula: $2E$ $\geq$ ...
Glo's user avatar
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0 answers
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Show that there exists a connected component $C'$ of $U' \cap G$ such that $C' \cap C \neq \varnothing.$

Let $G \subseteq \mathbb R^n$ be a domain and $z_0 \in \partial G.$ Let $U$ any connected open neighbourhood of $z_0$ and $C$ be a connected component of $U \cap G.$ Then there exists a connected open ...
Anacardium's user avatar
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1 answer
33 views

Simply connected set

An open connected set $S$ in $\mathbb{R}^2$ is said to be simply connected if its complement relative to the whole plane is connected. This definition is mentioned as an equivalent definition in the ...
Mathguide's user avatar
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2 answers
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Are connected components clopen?

I was thinking about this proposition from a book: "The components of $X$ are disjoint nonempty closed subsets whose union is $X$, and thus they form a partition of $X$" And I wondered: ...
some_math_guy's user avatar
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0 answers
51 views

Does Alexander duality hold without assuming local contractility for compacts on $\mathbb S^2$?

Consider $\mathbb S^2$ and two compact subsets $E, F \subset \mathbb R^2$ such that $F\subset E$. Additionally, assume that $F$ is a deformation retract of $E$. Question: Is it true that $\mathbb S^2 ...
Hugo's user avatar
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0 answers
17 views

Path existence between local maxima

Let $f:\mathbb{S}^2\to\mathbb{R}$ be a smooth function and let $M=\{x_1,\dots x_K\}$ be the set of its local maxima. I am trying to determine if there is a way to find out if there exists a '...
A P's user avatar
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1 vote
1 answer
79 views

After removing a point from a connected set in sphere, are the connected components still connected when the original point is added back

It can be written as theorem below (It may not true) If $X$ is a subset of $S^2$ and suppose $X$ is connected, $x_0$ is a point of X, we consider the set $X\setminus\{x_0\}$, if $X\setminus\{x_0\}$}=$\...
wxw030910's user avatar
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Lee Topological Manifolds Problem 4-12

I'm trying to prove the following statement: Problem 4-12: Suppose $X$ is a topological space and $S \subseteq X$ is a subset of $X$ which is both open and closed in $X$. Show that $S$ is a union of ...
Keshav Balwant Deoskar's user avatar
2 votes
1 answer
55 views

Every sequentially discrete space is totally path disconnected

A topological space $X$ is sequentially discrete if every convergent sequence in $X$ is eventually constant. A space $X$ is totally path disconnected if every path $f:[0,1]\to X$ is constant. It seems ...
PatrickR's user avatar
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0 votes
2 answers
77 views

Proving $S^2$ minus $2$ points are connected.

Prove $S^1, S^2$ are not homeomorphic. Assume there is a homeomorphism $f: S^1\to S^2.$ I have proved that if we remove a point $p$ in $S^1$, $S^1\setminus\{p\}$ is connected. I wonder if we remove a ...
user avatar
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0 answers
22 views

proving a space is connected [duplicate]

Let $X$ be a topological space. Prove that if $X=\bigcup\limits_{k=1}^{\infty}X_k$, $X_k$ is connected subset of $X$ and $X_k\cap X_{k+1}\neq\varnothing\quad\forall k\geq 1$, then $X$ is connected. I ...
user avatar
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0 answers
36 views

Irreducible representations of the identity component of the Spin group

Let $Spin(p,q)$ be the real spin group to the quadratic space $\mathbb{R}^{p,q}$ defined via Clifford algebras as $Pin(\mathbb{R}^{p,q}) \cap Cl^0(\mathbb{R}^{p,q})$, meaning as the subset of the ...
anonymous250's user avatar
4 votes
3 answers
119 views

If we remove the diagonal from $X\times X$, is it necessarily disconnected?

If $X$ is a compact, connected Hausdorff space, we know that the diagonal $\Delta_X=\{(x,x)\in X\times X\}$ is closed in $X\times X$ by Hausdorffness. But is $X\times X\setminus\Delta_X$ disconnected ...
tripaloski's user avatar
5 votes
0 answers
88 views

Removing an open curve from a regular, simply connected domain's boundary

This is a follow-up to my previous question. I now make my question more precise to avoid the counter-example given there. Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior ...
Desura's user avatar
  • 2,011
2 votes
1 answer
159 views

Is every subset of $X$ compact and connected under the trivial topology on $X$?

I was thinking about the trivial topology, in which the only open sets are $X$ and $\emptyset$, where $X$ is the entire space we are working in. My doubt is: are all the subset of $X$ compact and ...
Heidegger's user avatar
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