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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, simply connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

6
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1answer
83 views

Compact spaces in which any closed set can be partitioned into finitely many closed sets whose clopen subsets extend to the whole space

Let $X $ be a compact topological space (not necessarilly Hausdorff). I am looking for a charactrization for the following property: Property: If $C $ is a closed subset of $X $, then there are ...
1
vote
1answer
46 views

Connected component of $\{(x_1,…,x_n) \; | \; x_1^2+…+{x_{n-1}}^2-x_n^2 \neq 0\}$

The question is in the title, actually for $n=2$, it's okay, for $n=3$, I thought that : $\{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 < 0 \}, \{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 > 0 ...
-1
votes
1answer
22 views

Connected component and closed sets

how to prove this : Let $X$ be a topological space, such that $X=\displaystyle\bigcup_{ j\in J} F_j $ where $(F_j)$ is a family of nonempty closed disjoint connected sets. How to prove that $F_j$ ...
2
votes
2answers
46 views

Connected set in $C[0,1]$

Let $X=C([0,1])$ be the space of all continuous real valued function on $[0,1]$. Equip $X$ with the $\sup$ metric, $d(f,g)=\sup_{x\in[0,1]}|f(t)-g(t)|$ Let, $S=\{f\in X| f(0)\neq f(1)\}$ $T=\{f\in ...
2
votes
1answer
31 views

Proof that direct product of connected spaces is connected

I am trying to understand a piece of the proof that the direct product of connected spaces is itself a connected space, as given by Lee in "Introduction to Topological Manifolds". By induction, we may ...
1
vote
1answer
31 views

Where is my reasoning wrong?

We know that components of locally connected sets are open. Now consider the space $$X=\bigcup\limits_{q \in \mathbb{Q}}^{} \{(x,y) \in \mathbb{R}^2; x^2+y^2=q^2 \}-\{(0,0) \}$$ Now this seems locally ...
1
vote
0answers
22 views

connected subgraphs having minimum expansion

If I have a connected graph $G$, is it true that subgraphs(corresponding to cuts) having minimum conductance or expansion(or other connectivity measures) should be connected? In other words if $S$ is ...
1
vote
1answer
43 views

A path-connected graph is connected as a graph

This question regards the difference between being path-connected as a topological space, i.e. "Every two point $x,y \in X$ can be connected with a path" and being connected in a graph sense: "A graph ...
1
vote
1answer
44 views

IVT and connectedness on $\mathbb{R}^{n}$

Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be a continuous function and $n \geq 2$. Moreover, suppose that there is $c \in \mathbb{R}$ such that $f^{-1}(\{c\}) = \{x \in \mathbb{R}^{n} \mid f(x) = c\}$ is ...
2
votes
2answers
80 views

Assume $A \subset \mathbb{R}^n$ is connected such that $A^c$ is separated by $B,C$, then $A \cup B$ is connected.

The following problem showed up on a previous qualifying exam: Assume $A \subset \mathbb{R}^n$ is connected such that \begin{equation}A^c = B \cup C, \text{ such that } \overline{B} \cap C = \...
0
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0answers
18 views

Exact differential forms defined in path connected sets

There's a theorem in mathematics that states: Let $A$ be an open path connected set in $\mathbb{R^m}$, and $\omega:A\to\mathbb{R^{1\times2}},x\to[\omega^1(x)\space...\space \omega^m]$ be a 1 degree ...
0
votes
0answers
59 views

Topologist Sine Curve, connected but not path connected

So I have to show that $X=U∪V$, $U=\{(0,y),y\in[-1,1]\}$, $V=\{(x,sin({1 \over x}),x>0\}$ is connected but not path connected My proof: Assume that $X $ is path connected and let $f:[0,1]\...
0
votes
1answer
19 views

Connected space and his relathionship with his own subset

I saw this proposition without the prove and I thought that it was wrong. $X$ connected if, and only if, any own subset $A$ from $X$ has some point of his border.In other words, his border isn´t ...
0
votes
1answer
12 views

The union of path-connected sets sharing a nonempty path-connected set $K$, is path-connected?

If a metric space $M$ is the union of path-connected sets $S_{\alpha}$, all of which have the nonempty path-connected set $K$ in common, is $M$ path-connected? I didnt have a good idea. I know that a ...
0
votes
1answer
45 views

Show that the closure of a connected set is connected.

Show that if $A$ is a connected subset of a metric space $X$, then $\overline{A}$ is connected. My approach: If $A$ is a connected subset of metric space $X$, then $\emptyset$ and $A$ are open and ...
1
vote
0answers
59 views

When a subspace of a space is totally disconnected

Let $X $ be a topological space and $Y $ a Hausdorff subspace of $X $ such that for every connected component $C $ of $X $ the set $C\cap Y $ is finite. How can we show that $Y $ is totally ...
0
votes
3answers
70 views

How can I prove that $\mathbb Q \times \mathbb Q$ is connected?

How can I prove that $\mathbb Q \times \mathbb Q$ is connected? I was trying to prove that it isn't because $\mathbb Q$ is not connected but I can't find two open $A,B$ to prove it.
1
vote
1answer
26 views

A connected embedded submanifold, which is contained in an immersed submanifold, is connected in this submanifold?

I'm currently studying foliation theory, and to solve a problem I need the following to be true, but I can't prove nor disprove, I just have a feeling it may be false in general: if I have a smooth ...
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0answers
19 views
0
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2answers
31 views

Is every nonopen connected subset of the circle path-connected?

(a) Is every open connected subset of the circle path-connected? (b) What about connected nonopen subsets of the circle? For (a), the circle is a subset of $\mathbb{R}^{2}$ and if $U$ is an open ...
0
votes
3answers
65 views

Show that if $f(-1)<0$ and $f(1)>0$, then there exist $-1<a<1$ such that $f(a)=0$

Let $f:\mathbb{R}\to\mathbb{R}$ continuous. Show that if $f(-1)<0$ and $f(1)>0$, then there exist $-1<a<1$ such that $f(a)=0$. My approach: Suppose that $f(x)\neq 0$ for all $x\in (-1,1)$....
2
votes
1answer
59 views

If $f: M \to \mathbb{R}$ is a continuous function such that all values are irrationals, then $f$ is constant whenever $M$ is connected?

(a) Prove if $f: M \to \mathbb{R}$ is a continuous function such that all values are integers, then $f$ is constant whenever $M$ is connected (b) What if all values are irrationals? My attempt. ...
1
vote
1answer
33 views

The interior of a disconnected set is disconnected?

(a) The closure of a disconnected set is disconnected? (b) What about the interior of a disconnected set? For (a) take $\mathbb{R}^{2}-\{x\text{-axis}\}$. For (b) seems true. I just had an ...
1
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0answers
36 views

$|f(z)|=c\in\Bbb R\forall z\in\Omega$ for $\Omega\subset\Bbb C$ open and connected $\implies f$ is constant. Why must $\Omega$ be open, connected?

Let $f$ be holomorphic on $\Omega\subset\Bbb C$, and $\Omega$ is open and connected. If $|f(z)|$ is constant it is known that $f$ is constant on $\Omega$ The proof of this fact says that if $f$ is ...
2
votes
1answer
31 views

Is the following exercise on connected subset wrong?

I think there is a problem with the following exercise from Zorich, Mathematical Analysis II - Exercise 1 in Section 9.4.1: Show that in terms of the ambient space the property of connectedness of ...
1
vote
0answers
100 views

Queris related to “An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$.”

The following proof of An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$ is provided by my professor.I've some ...
1
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1answer
75 views

Interesting Graph Problem?

I am given an undirected connected graph of $N$ vertices and $M$ edges. I need to make this undirected graph directed by directing edges.I need to check if it's possible to make a directed graph such ...
1
vote
1answer
35 views

Infinite “connected” graphs and spanning trees

Let us say that graph $G$ is quasi-connected iff for every two vertices there is either a finite path or an infinite path in sense of the offtopic of this answer. Assuming the axiom of choice, is it ...
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votes
2answers
50 views

Why is my proof that $\mathbb R$ is disconnected wrong?

The definition of connectedness in my notes is: A topological space $X$ is connected if there does not exist a pair of non empty subsets $U$, $V$ such that $U\cap V=\emptyset$ and $U\cup V=X$. ...
2
votes
1answer
27 views

A cover of Locally connected space with certain compactness property

Suppose $X$ is a locally connected Hausdorff space. If $X$ is $\sigma$-compact and locally compact, is it always possible to find a countable set of precompact connected open sets $\{U_n\}$ (which ...
0
votes
2answers
38 views

Counterexamples about function discontinuity.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function with a point $\textbf{x}\in\mathbb{R}^n$ of discontinuity. Is it possible that the image $f(O_{x_i})$, the image of an open ball (containing $...
0
votes
1answer
35 views
1
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0answers
57 views

Property of set $\left \{y \in \Bbb R : y =\lim\limits_{n \rightarrow \infty} f(x_n), \text {for some sequence }\ x_n \rightarrow +\infty \right \}$ [closed]

Let $f : \Bbb R \longrightarrow \Bbb R$ be a continuous function and $A \subseteq \Bbb R$ be defined by $$A=\left \{y \in \Bbb R : y =\lim_{n \rightarrow \infty} f(x_n), \text {for some ...
1
vote
1answer
27 views

A connected maximal graph $G$ with no cycles of length at least $k+1$ has $|V(G)| \leq k$ or has a cut-vertex when $k \geq 2$

Is the following true: A maximal graph $G$ with no cycles of length at least $k+1$ has $|V(G)| \leq k$ or has a cut-vertex when $k \geq 2$. Here maximal is taken to mean that no edge can be added ...
1
vote
2answers
38 views

Does exist a compact connected $K' \subset U$ such that $K \subset K'$, if $U$ is an open connected and $K$ in $U$ a compact?

Let $U$ be an open and connected set in $\mathbb{R}^n$. Suppose $K \subset U$ is a compact set. Is it true that there exists a compact and connected set $K' \subset U$ such that $K \subset K'$? I ...
1
vote
0answers
26 views

Showing a subset of $\mathbb{C}$ is simply connected via homotopy

This question is Exercise 19 in Chpater 8 of Stein and Shakarchi's Complex Analysis. Prove that the complex plane slit along the union of rays $\cup_{k=1}^{n}\{A_k+iy:y\leq 0\}, A_k \in \mathbb{R}$ ...
0
votes
4answers
184 views

Can an undirected graph be disconnected?

This may be a rather trivial question but I am still trying to get the hang of all the graph theory terms. Nonetheless, I haven't found a source that explicitly says that an undirected graph can only ...
2
votes
1answer
18 views

What is the maximum connectivity of a planar graph?

The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph? In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is ...
0
votes
0answers
14 views

Prove a family of connected sets with one set intersecting all others is connected [duplicate]

The question: If $\{A_j : j \in S\}$ is a family of connected sets and if one set of the family, $A$ intersects all the others, prove that $X = \cup_{j \in S} A_j$ is connected. My attempt at the ...
0
votes
1answer
52 views

A graph (G) is only connected as a topological if G is connected as a graph. [duplicate]

So in our textbook it's stated that "A Graph G is connected as a topological space if and only if G is connected as a graph" without a proof and my lecturer did not prove it during the lecture, so I ...
1
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0answers
10 views

Reduction of Steiner Tree to Maximum Weight Connected Subgraph

I want to proof that the MWCS is an NP-hard problem by proofing that its decision version is NP-Complete. Below I have my proof so far and I hope people can give comments whether it can be formulated ...
0
votes
1answer
21 views

Maximum possible connected component by removing hyperplanes form $\mathbb R^3$

Let $A_1,A_2,A_3,A_4$ be 4 hyperplanes in $\mathbb R^3$ then How many maximum possible connected component are present in $\mathbb R^3$ after removing these 4 hyperplane I can visuallise that if i ...
2
votes
2answers
44 views

Is the set $\{2,3,5\}$ connected?

I read somewhere that a connected set in the digital line topology is a subset of consecutive integers. Recall that the digital line topology is a topology on $\mathbb{Z}$ with basis elements $\{n\}$ ...
3
votes
2answers
51 views

Determine whether $S=\{x,y\in \mathbb{R}^2 : x^2-y^2=1\}$ is disconnected or connected.

So I want to prove its disconnected. That is $S_1\cup S_2= S$ for $S_1,S_2$ being non empty, disjoint relatively open sets. My proposed sets are $S_1=\{x,y\in \mathbb{R}^2 : x>0, x^2-y^2=1\}$ and $...
3
votes
2answers
76 views

Connected but not path-connected

Let $r:[0,\infty)\rightarrow\mathbb{R}$ be given by $r(t)=1+e^{-t}$. Let $S\subset \mathbb{C}$ be the image of the "spiral" curve $f:[0,\infty)\rightarrow\mathbb{C}$ given by $f(t)=r(t)e^{it}$. Then $\...
2
votes
2answers
143 views

Path-conectedness of open balls implies path-connectedness

Let $M$ be a subset of a metric space $(E,d)$. I have just proved that $M$ path-connectedness implies $M$ connectedness. Now I need to show that if $M$ is connected and every open ball in $E$ is path-...
2
votes
2answers
46 views

$U\subseteq\mathbb{R}^n$ is open and connected $f:U\to \mathbb{R}^m$ differentiable.$Df(x)=0$ $\forall x \in U$. then $f$ is constant.

Attempt: Since U is open and connected, U is path connected. then there exists a path $\gamma(t):[0,1]\to U$ between any points $a,b\in U$ such that $\gamma(1)=b$ and $\gamma(0)=a$ Define $h(t)=(f\...
3
votes
1answer
56 views

Confusion about the proof of Menger's Theorem in “Introduction to Graph Theory” by Douglas West

The proof of Menger's Theorem in the book "Introduction to Graph Theory" by Douglas West (2nd Edition; Page 167) has been divided into two cases. The second case assumes that "Every minimum $x,y$...
0
votes
2answers
57 views

Manifolds of zero dimension and $\mathbb R^0$?

Tu Manifolds Section 5.4 Example 5.13 (Manifolds of dimension zero). In a manifold of dimension zero, every singleton subset is homeomorphic to $\mathbb R^0$ and so is open. Thus, a zero-...
2
votes
1answer
24 views

Proof that a singleton set must be closed or open when its ambient space is connected and given subspace is disconnected

I have been asked to prove the following: Let $X$ be a connected topological space, let $p\in X$, and let $X-\{p\}$ be disconnected. Prove that $\{p\}$ must be either open or closed in $X$, but ...