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.

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24 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 ...
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1answer
23 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
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1answer
26 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 ...
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2answers
78 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
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2answers
77 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 ...
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1answer
11 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
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1answer
50 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 ...
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2answers
25 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
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1answer
35 views

Can the union of disjoint open sets be connected? [closed]

Let $U$,$V$ open sets, such that $U\cap V =\emptyset $. Can $U\cup V$ be connected? If not, how do I prove it?
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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 ...
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2answers
33 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 ...
2
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0answers
31 views

Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...
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3answers
51 views

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ is path-connected.

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ defined by $U_\epsilon(A) := \{x \in \Bbb R^n : d_A(x) < e\}$ is path-connected. If ...
3
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2answers
67 views

Every local property for $\mathbb{R}$ (any Connected Separable Space) holds globally?

I'm given this problem : Prove that "being polynomial" is a local property, meaning if $f: ℝ → ℝ$ is a polynomial in a neighborhood of each real point, then $f$ is a polynomial. I think I ...
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2answers
29 views

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected.

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected. Here I use the following criterion for $X$ to be connected: A metric space $(X,d)$ is ...
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1answer
21 views

$f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , is the graph of $f$ connected in $\mathbb R^2$?

Consider the function $f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , then $f$ is not continuous on $\mathbb R$ . Is the graph of $f$ i.e. $G(f) :=\{ ...
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2answers
46 views

A connected space that admits a nonconstant continuous map into reals is uncountable

Let $f:X\to Y$ be a non-constant continuous map of topological spaces. If $Y=\mathbb{R}$ and $X$ is connected then $X$ is uncountable. True or False? I know that $f(X)$ is an interval. The ...
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2answers
32 views

Let $U$ be an open connected subset and $f : U \to \Bbb R$ be a diff function then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$ then $f$ is a constant function. If we can prove that ...
6
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2answers
67 views

Prove that $ (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally connected and connected subspace of $\mathbb{R}^2$

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that $A = (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally ...
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1answer
31 views

Equivalent Definition of disconnectedness of a Metric Space X

I'm looking for a proof of this theorem, which states the equivalent definition for disconnectedness of a metric space X. Especially, I'm looking for a proof of (1) <=> (2)! does anyone can ...
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0answers
18 views

$\cup \mathcal C$ is polygonally connected.

Suppose $X$ is a normed linear space and $\mathcal C$ is a chained collection of convex subsets of $X$ then $\cup \mathcal C$ is polygonally connected. A non-empty collection $C$ of sets is said to ...
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0answers
52 views

If $E= A\cup B \cup C$ and $E$ is connected , where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected.

If $E= A\cup B \cup C$ and $E$ is connected in a metric space $(X,d)$, where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected. If we consider that $A \cup C$ is not ...
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1answer
21 views

Proving a graph is connected

I have to prove that if a graph G=(V,E) has |V|=2n and every vertex has a degree of n, (with n<-N*) then it is connected. I have this so far: If there are 2n verticles and each one has a degree ...
1
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1answer
39 views

Is it path-connected space?

I have a finite topological space $X= \{ 1,...n \}$ with the following topology that I created $T=\{ \emptyset, X \} \bigcup \{A \subseteq X | 1\in A\} $ It is connected because $\bigcap_{A \in X, ...
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1answer
38 views

Maximum edge of a directed graph , if it contains weak components?

A digraph includes n nodes , and has two weak components , what is the maximum number of edges? ( there is no directed cycle)? Another question ,how does the answer change , if there is two strong ...
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3answers
88 views

Showing that arcs do not separate the plane $\mathbb{R}^2$

Is $\mathbb{R}^2\setminus f([0,1])$ connected if $f:[0,1]\to\mathbb{R}^2$ is an embedding? It seems that this is clearly true but I am having a hard time proving it. The only things that I know is ...
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2answers
43 views

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$.

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$. We assume on the contrary that there does not exist ...
1
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1answer
72 views

$f(X)$ is uncountable and hence $X$ is uncountable.

My question: let $f : X \to \Bbb R$ be a non-constant continuous function on a connected metric space and assume that $f(X)$ is uncountable; then $X$ is uncountable. We know continuous image of a ...
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2answers
39 views

Non-Hausdorff space such that all connected components are singletons

Is there a topological space $(X,\tau)$ such that $(X,\tau)$ is not Hausdorff; if $S\subseteq X$ and $S$ contains more than 1 point, then $S$ is not connected (with the subspace topology).
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1answer
44 views

$A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected

Let $X$ be a topological space. Let $A$ and $B$ be two subsets of $X$ such that: $A$ and $B$ are closed, $A \cup B$ is connected, $A \cap B$ is connected Prove that $A$ and $B$ are connected. ...
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4answers
87 views

There's no continuous injection from the unit circle to $\mathbb R$

I read a proof that goes as follows: Let $U$ be the unit circle, and let $f : U \longrightarrow \mathbb R$ be a continuous mapping. $U$ is compact and connected, so $f(U)$ is a closed, bounded ...
2
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1answer
50 views

Connectedness of $S^2$

I'm taking a real analysis course, as I've said before, and the professor has been teaching a lot of topology. We don't have a textbook, so I have to use his notes which are very confusing. One ...
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1answer
25 views

$\{U_{\alpha} \}_{\alpha \in I}$ collection of connected sets , for every $U_{\alpha}$ , $\exists U_{\beta}\ne U_{\alpha}$ not mutually disjoint

A probable further strengthening of $\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint ... If $\{U_{\alpha} \}_{\alpha ...
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2answers
32 views

$\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint …

If $\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint , then is the union of all the sets in the collection connected ...
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2answers
42 views

Suppose that $A$ is a connected subset of a space $X$ and that $A\subseteq B \subseteq \bar A$. Prove $B$ is connected. [duplicate]

I think I can prove the closure of $A$, that is $\bar A$, is connected, as there are many other threads on this site. I am then just not sure how to make the jump to show formally that B is ...
1
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0answers
19 views

Let $V$ be a NLS (over $\mathbb R$ ) of dimension $>1$, then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected?

Let $V$ be a normed linear space (over $\mathbb R$ ) , then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected ? I know that if $V$ is the space of complex numbers ...
2
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1answer
37 views

To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected.

To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected. I want to do it without path-connectedness or polygon-connectedness using the fact continuous image of a connected ...
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1answer
37 views

$(\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\{0\})$ is path-connected but not localy connected

I have to prove that $A=(\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\{0\})$ is path-connected, one in the chat suggested to take $$\varphi(t)=\begin{cases} (x,(1-3t)y), t\in [0,\frac13]\\ ...
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1answer
33 views

The set $\{(x,\sin(\frac1x), x\in]0,1]\}\cup (\{0\}\times[-1,1])$ is not path-connected

As in the titel i try to prove that $$B=\{(x,\sin(\frac1x), x\in]0,1]\}\cup (\{0\}\times[-1,1])=A\cup F$$ is not path-connected I suppose by contradiction that $B$ is path-coonected, so there exist ...
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0answers
19 views

How to show $\mathbb{R^2}$ is sequentially connected without path-connectedness

Definitions: Connected: Not separated Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and ...
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1answer
74 views

$\{\frac1n, n\in \mathbb{N}^*\}$ is localy connected

I have this set $A=\{\frac1n, n\in \mathbb{N}^*\}$ How can I prove that this set is locally connected but $\overline{A}$ is not locally connected. Thank you
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1answer
35 views

If every An is connected, then A is connected

I'm practicing for a midterm, so I've been finding exams with answer keys online and solving through them. The last question of this one reads as follows: Let $R^2$ be given the usual metric ...
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2answers
36 views

Connectedness of $\{(x,\sin(\frac{1}{x})); x \in ]0,1]\}$

Let $A = \{(x,\sin(\frac{1}{x})); x \in ]0,1]\}$ I need to show that $A$ is connected. I am trying to use the following theorem: If $(X,d_1)$ and $(Y,d_2)$ are two metric spaces, and $f: X ...
1
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1answer
37 views

Paths and connectivity of graphs

I am trying to show that for a graph on $n\ge 3$ vertices with minimum degree of all vertices $\ge k/2$, G connected that G has a path of length k. I know if n is greater than k but n/2 is less than ...
4
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4answers
77 views

Is it a connected set in $\mathbb{R}^2$?

I have this set $B=A_1\cup A_2\cup A_3\cup \{(0,0)\}$ where $A_1=\{(x,y)\in \mathbb{R}^2, y>x^2\}\\A_2=\{(x,y)\in \mathbb{R}^2, 0<y<x^2, x<0\}\\A_3=\{(x,y)\in \mathbb{R}^2, ...
1
vote
1answer
44 views

Proof: “Each vertex belongs to exactly one connected component”

On this Wikipedia page: http://en.wikipedia.org/wiki/Connectivity_%28graph_theory%29 they state "Each vertex belongs to exactly one connected component". How can this be proven formally?
4
votes
1answer
58 views

Is the following set (path) connected?

This is a homework question. $d,n\ge 2$. Let $L=\{(x_1,...,x_n)\in (\mathbb{R}^d)^n: x_i\in \mathbb{R}^d, x_i\ne x_j \forall i\ne j\}$. I tend to think it is not path connected because if you ...
0
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0answers
16 views

Prove if $\{(x,y) \in \mathbb{R^2} : 0\leq x\leq 1, 0\leq y\leq 1\}$ is separated, then there is a separation of $0 \leq z \leq 1$

Definition of separated: Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have ...
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3answers
73 views

The annulus is connected without using path-connectedness [closed]

How to prove that this set $$\{z\in \mathbb{C}, 1<|z|<2\}$$ is connected ? Thank you.
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0answers
44 views

Every Hilbert space is connected

Let $H$ be a Hilbert space. Proving $H$ is connected, suppose $\{e_i\}_{i\in I}$ is a orthogonal basis of $H$. Thus $H=\bigoplus_{i\in I} \Bbb C e_i$. Clearly $\Bbb Ce_i$ is connected for every $i$, ...