Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, simply connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

learn more… | top users | synonyms

-2
votes
1answer
21 views

Proof cube & trapezium a Compact Space & $E^n$ & $I^n, I^{\infty}$ are connected space ??? [on hold]

I need a serious help here please! Question 1: Prove that $E^n$ & $I^n$, $I^{\infty}$ are connected spaces. After a lot of search I found some two theorems in James Dugundji book. But I still ...
0
votes
1answer
34 views

Fundamental groups in path-connected space

I'm studying Fundamental groups and today I saw the follow theorem: Theorem: Let be $X$ a topological space path-connected and $x,y\in X$. Then, the application $\psi:\pi_1(X,x)\to \pi_1(X,y)$ is a ...
0
votes
1answer
37 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
1
vote
0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
0
votes
1answer
24 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
0
votes
0answers
7 views

Line integrals in a double connected set

If P and Q are continuously differentiable on an open doubly connected(one hole) region $R$, and if $\partial P/\partial y = \partial Q/\partial x$ everywhere in $R$, how many distinct values are ...
2
votes
0answers
60 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...
3
votes
1answer
60 views

Is it true that the intersection of a sequence $K_1 \supset K_2 \supset K_3 \dotsm$ of connected subsets of $\mathbb{R}^2$ is also connected?

I have got one counterexample for this : Consider the family {D} of closed discs centered at zero having radius $1+1/n$, i.e. disc $D_1$ has radius $1+1=2$, $D_2$ has radius $1+1/2=1.5$, and so on. ...
0
votes
1answer
30 views

Let $K_1 \supset K_2 \supset… $ be a sequence of connected compact subsets of $ \Bbb R^2 $. Is $ K = \cap_{i=1}^\infty K_i $ is connected? [duplicate]

I have managed to write down two proofs showing the connectedness of $K$. But still shaky about both of them. Here are the proofs: 1)Suppose $K$ is disconnected. Then we write it's separation as $ K ...
0
votes
1answer
62 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
0
votes
0answers
34 views

Boundary disconnects connected space

I need to show that a boundary of an open subset $\emptyset \neq A\subset X$ whis isn't dense in a connected space X, disconnects X. Any hints how should I start?
-1
votes
0answers
18 views

decomposition of a 3-regular connected graph [closed]

I want to prove that every 3-regular connected graph can be decomposed into two sub graphs with connected component of less than 3.
2
votes
1answer
54 views

Between any two vertices $u,v$ in a 3-connected graph, there are two internally disjoint $u$-$v$ paths of different lengths?

I am trying to solve the following exercise about 3-connected graphs from this book. (a) Show that for every two vertices $u$ and $v$ of a $3$-connected graph $G$, there exist two internally ...
2
votes
1answer
52 views

Prove that $\partial A$ is a cutset of connected $X$ if $\operatorname{Int}(A)$ and $\operatorname{Int}(X - A)$ are nonempty

Exercise 6.23 (p.202) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa asks: Let $X$ be a connected topological space and $A$ be a subset of $X$. Prove that if ...
1
vote
1answer
20 views

Finding $k$-clique in a graph with running time of $|V|^{k-1}$

This is a homework problem. Let's say I have a graph $G$, how can I find a $k$-clique (i.e. a complete graph with $k$ vertices) inside $G$? So far I can think of a naive solution where I check if each ...
3
votes
1answer
35 views

Proving a corollary to the Jordan Curve Theorem

A Jordan curve is a continuous closed curve in $\Bbb R^2$ which is simple, i.e. has no self-intersections. The Jordan curve theorem states that the complement of any Jordan curve has two connected ...
0
votes
0answers
17 views

Is the collection of matrices with rank at most k forms a connected subset of the space of matrices?

Let $M_{m\times n}$ be the space of $m\times n$ complex matrices and let $1\leq k\leq \min\{m,n\}$. Also, let $R_k=\{A\in M_{m\times n} : \|A\|_F=1, \mathrm{rank}(A)\leq k\}$, where $\|\cdot\|_F$ ...
4
votes
2answers
49 views

Let $A$ be a closed subset of a connected top space $X$. If the $\text{Bd }A$ is connected, does it imply $A$ is also connected?

I came across a question from Munkres: Problem 24.11: If A is a connected subspace of X, does it follow that IntA and BdA are connected? Does the converse hold? Thinking about the converse, I know ...
1
vote
1answer
21 views

Prob. 10 (b), Sec. 25 in Munkres' TOPOLOGY, 2nd ed: How to show that components and quasicomponents are the same for locally connected spaces?

Let $X$ be a topological space; let us define $x \sim y$ if there is no separation $X = A \cup B$ of $X$ into disjoint open sets such that $x \in A$ and $y \in B$. This relation is an equivalence ...
1
vote
0answers
25 views

Prob. 5, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is there a connected set that is locally connected at none of its points?

Let $A$ denote the rational points of the interval $[0,1] \times 0$ of $\mathbb{R}^2$. Let $T$ denote the union of all line segments joining the point $p = 0 \times 1$ to points of $A$. Then I can ...
1
vote
1answer
38 views

Prove that the complement of an open ball in $\mathbb{R^n}$ has exactly one unbounded component [duplicate]

Question: Let $B^n \subset \mathbb{R^n}$ be open ball in the Euclidean metric. Prove that the complement of $B^n$ in $\mathbb{R^n}$ has exactly one unbounded component (components of a set are class ...
-1
votes
1answer
49 views

Prob. 3, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is $I \times I$ path connected or locally path connected in the subspace topology?

Let $$I \ = \ [0,1] \ = \ \{\ x \ \in \mathbb{R} \ \colon \ 0 \leq x \leq 1 \ \}. $$ In the subspace topology that $I \times I$ inherits from the dictionary order topology on $\mathbb{R} \times ...
0
votes
0answers
25 views

Is either of these subsets connected in $\mathbb{R}^\omega$ in the product, uniform, box topologies?

Let $\mathbb{R}^\omega$ be the set of all (infinite) sequences of real numbers. Let $A$ be the set of all bounded sequences of real numbers, and let $B$ be the set of all unbounded sequences. Then ...
0
votes
1answer
18 views

Prob. 4, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is every connected subset of a locally path connected space not path connected?

Let $X$ be a locally path connected topological space. Let $A$ be a non-empty connected subset of $X$. Then $A$ is path connected. Proof: Let $C$ be the component of $X$ that intersects $A$. Then ...
0
votes
1answer
26 views

Probs. 2 (b) and 2 (c), Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Components in the uniform and box topologies

Let $X$ be a topological space. Then, for any points $x, y \in X$, we can define a relation $\sim$ on $X$ by defining $x \sim y$ iff there is a connected subspace of $X$ containing both $x$ and $y$. ...
0
votes
0answers
18 views

every nonempty compact, locally path-connected and connected metric space is path-connected [duplicate]

I wanna prove that if $M$ is nonempty compact, locally path-connected and connected metric space then it is path connected. I think to prove this the best way is to show that between every to points ...
2
votes
1answer
59 views

the topologist's sine circle is path-connected but it's not locally path-connected

As you may know the topologist's sine curve is the set: $\{{(x,y) : x=0 \ and \ |y|\leq 1,\ or \ 0<x \leq 1 \ and\ y=\sin\dfrac{1}{x}}\}$ I want to show that the topologist's sine circle which is ...
0
votes
0answers
31 views

Prob. 12, Sec. 23 in Munkres' TOPOLOGY, 2nd ed: How are these unions connected?

Let $Y \subset X$; let $X$ and $Y$ be connected. Suppose that $A$ and $B$ form a separation of $X - Y$. Then how to show that $Y \cup A$ and $Y \cup B$ are connected? My effort: Now since $A$ ...
1
vote
1answer
44 views

Example 2, Sec. 25 in Munkres TOPOLOGY 2nd ed: Is this subspace also connected?

The topologist's sine curve is the closure $\overline{S}$ of the subset $S$ of $\mathbb{R}^2$, where $$S\ = \ \{ \ x \times \sin \frac{1}{x} \ \colon \ 0 < x \leq 1 \ \}.$$ So $$\overline{S} = S ...
2
votes
1answer
58 views

Example 3, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Why is the topologist's sine curve not locally connected?

The topologist's sine curve is by definition the closure $\overline{S}$ in $\mathbb{R}^2$ of the set $S$ given by $$S = \{ \ x \times \sin \frac{1}{x} \ \colon \ 0 < x \leq 1 \ \}.$$ Let $V = \{ ...
3
votes
2answers
44 views

Connectivity of a subset of the topologist's sine curve

I have a question about Example 2 of Section 25 (p.160) of Munkres's Topology. Let $S$ be the following subset of the plane $\mathbb{R}^2$: $$S = \{ \ x \times \sin \tfrac{1}{x} \ \colon \ 0 < x ...
0
votes
0answers
30 views

Prob. 9, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to achieve this? [duplicate]

Let $A$ be a countable subset of $\mathbb{R}^2$. How to show that $\mathbb{R}^2 - A$ is path-connected? My effort: Let $A = \{ a_1, a_2, a_3, \ldots \}$, where $a_n = (\alpha_n, \beta_n) \in ...
3
votes
4answers
316 views

Prove that the graph is connected

I was wondering if someone can help me understand how prove that this graph is connected. Given a graph with n vertices, prove that if the degree of each vertex is at least $(n − 1)/2$ then the graph ...
2
votes
1answer
46 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
2
votes
1answer
46 views

Connected subset

I have the follwing question: Let $M$ be a connected metric space, $X \subset M$ is connected. Show that, if $A \subset M - X$ is both open and closed in $M - X$ then $A \cup X$ is connected. I don't ...
3
votes
2answers
55 views

Preserving compactness and connectedness implies continuity for functions between locally connected, locally compact spaces?

In this question: Connected and Compact preserving function is not continuous example? It is mentioned that "a function between locally-compact, locally-connected topological spaces which preserves ...
0
votes
1answer
46 views

In a connected graph, if the maximum path you could make is of length 100, and there are two paths of length 100, aren't they the same path?

Here's the question: Let G be a connected graph. (Remember that this means that every two vertices of G can be joined by a path starting at one and ending at the other.) Suppose also that G ...
-2
votes
2answers
27 views

The image of a path-connected set under a continuous map is path-connected

Show that if $X$ is path-connected and $f:X\to Y$ is a continuous map, then the image $f(X)$ is path-connected. In order to show this is path connected I know the definition is :
0
votes
0answers
25 views

Let $X_1$ and $X_2$ be connected topological spaces. I want to show that the product $X_1 \times X_2$ is connected. [duplicate]

Let $X_1$ and $X_2$ be connected topological spaces. I want to show that the product $X_1 \times X_2$ is connected. By definition the base from which the topology $\mathcal J_{X_1 \times X_2} = ...
0
votes
1answer
14 views

How to show $[0, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, 2\pi)$ is sequentially separated?

Definition of separated: $E$ is separated if $A, B \neq \varnothing$, $A \cup B = E$, and there are not convergent sequences in $A$ that have limit points in $B$, and vice versa. Definition of ...
0
votes
0answers
25 views

Proving that $S^n$ (n-sphere) is locally connected.

Definition: A space X is said to be locally connected at $x\in X$ if for any open set $U$ containing $x$, there is an open connected subset of $U$ (say $W$) containing $x$.$$x\in W\subseteq U$$ A ...
1
vote
1answer
26 views

What is the condition for Morera's theorem to be true?

The answer could be chosen from a) simply connected domain b)connected domain c)no conditions(true for any complex domain) I chose c because the theorem(in our textbook, at least) does not imply ...
2
votes
1answer
28 views

About connected topological subgroup

I'm trying to understand a proof of a theorem but I didn't understand a point. Let $G$ be an locally compact abelian group. Denote $G_0$ the connected component of $0$ (the identity of $G$). It's an ...
10
votes
2answers
100 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
2
votes
2answers
96 views

In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Prove that if $X$ is Hausdorff and $\mathfrak{C}$ is a nonempty chain of compact and connected subsets of $X$, then $\bigcap \mathfrak{C}$ is compact and connected. Here are the definitions which ...
0
votes
1answer
14 views

Asymptotic convergence of the total length of a graph

I encoded the following algorithm: suppose we're in (0,1)x(0,1) and I randomly create a "village" one at a time. At each step, I link a newly randomly created village to the closest village already ...
2
votes
1answer
52 views

Proving that $H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$

In the article about the Reduced Homology it's stated that $$H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$$ but I don't know how to prove that. I know $$H_0(X)=\bigoplus_{\alpha\in ...
0
votes
2answers
28 views

If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$.

Let $(X,d)$ be a metric space and $A,B \subset X$. If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$. I tried it with proving a contradiction if we first ...
1
vote
1answer
19 views

Let $X$ be the union of axes is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?

Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$? If we remove the origin from the union of axes ...
1
vote
2answers
35 views

Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0 $ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...