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.
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Linear continuum of $I\times I $ under subspace topology of $\mathbb{R}^2$ with dictionary topology on it, where $I=[a,b]$.
Let $I\times I$ be subspace of space $\mathbb{R}^2$ with dictionary order, where $I=[a,b]$. What can you say about the linear continuum of $I\times I$ with the subspace topology.
{ I've proved that ...
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1answer
16 views
Continuously rotating a unit vector to $e_1$
The following question is from Brian C. Hall's Lie Groups, Lie Algebras,
and Representations.
Show that $\mathrm{SO}(n) $ is connected, using the following outline.
For the
case $n =1$, ...
0
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1answer
22 views
Union of connected sets with possible empty intersection
There is a result which states that if a collection $A$ of connected sets has a point $P$ belonging to every of those sets, then its union is connected
I was wondering if this remains true if the ...
1
vote
1answer
46 views
Contractable Metric Spaces Homeomorphic to Euclidean Space
Is there a characterization of all metric spaces which are homeomorphic to a contractable subset of Euclidean space?
This question is cross-referenced here.
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2answers
53 views
Complement of unit sphere is disconnected?
I have proved that unit sphere is connected in $\Bbb R^3$, can I use this fact to prove that complement of unit sphere is not connected?
0
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1answer
32 views
Moore plane and connected spaces(Topology) [closed]
¿Is the Moore plane a connected space?
I have tried to find a counterexample but i can't.
1
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1answer
34 views
Open connected which does not contain open convex subsets
It is well known that convex implies connected, and it is clear that if $X$ is a locally convex topological vector space and $\emptyset \neq A\subseteq X$ is open then $A$ contains a nonempty open ...
0
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0answers
23 views
Arc connectedness implies connected
I've seen that are one or two questions like this, but I'm not fully convinced that are right. Both pretty much say the same thing, so I leave the easier to understand:
path connectedness implies ...
0
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0answers
17 views
Find the minimum positive integer $r$ for which there exists an $r$-regular graph $G$ such that $\kappa(G) \neq \lambda(G)$
Find the minimum positive integer $r$ for which there exists an $r$-regular graph $G$ such that $\kappa(G) \neq \lambda(G)$. Verify your answer.
(Chartrand, Gary, and Ping Zhang. A First Course in ...
4
votes
1answer
138 views
Existence of a symmetric subset $B\subseteq A$ such that $2A-A\subseteq 8A$
Let $A$ be a nonempty open connected subset of a (real) topological vector space $X$ such that $$2A-A \subseteq 8A$$ (for instance one could take $A=(-1,2)$).
Question. Is it true that there exists ...
0
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1answer
14 views
Existence of symmetric subsets
Let $A$ be a nonempty open connected subset of a (real) topological vector space $X$.
Question. Is it true that there exists a nonempty open connected set $B\subseteq A$ such that $B$, in addition,...
0
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1answer
76 views
What is the equivalent of a tree for directed graphs?
A tree is defined as a connected acyclic undirected graph at page 171 of this online book.
What is the equivalent of a tree for directed graphs? A connected acyclic directed graph (i.e. a connected ...
2
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1answer
38 views
The subspace of $M_{n\times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected
Let $m$ and $n$ be positive integers and $0\le k \le min\{m,n\}$ an integer . Prove or disprove : The subspace of $M_{m\times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is ...
0
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0answers
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Find the connected component of given set.
$Let \;𝑋$ = $\{(𝑥, 𝑦) ∈ ℝ^2: 𝑥^2 + 𝑦^2 = 1\} \bigcup([−1,1] × {0}) ∪ ({0} × [−1,1]).$
Let$\; 𝑛_0$ = $\max\{𝑘 ∶ 𝑘 < ∞,$ there are$\; 𝑘$ distinct points$\; 𝑝_1, … , 𝑝_𝑘 ∈ 𝑋$ such that $�...
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0answers
37 views
Path-Connected Components [duplicate]
How many path-connected components does $O_{2}(\mathbb{R})$ have?
My Attempt: I believe that the answer is 2, but I'm not sure how to work with the definition and describing the path-connected ...
2
votes
1answer
21 views
Connected metric space of first category
Question:
Does there exist a connected metric space of first category (i.e. it can be written as a countable union of nowhere dense subsets)?
No such example is given in the book Counterexamples ...
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0answers
12 views
Connectivity property between k and k+1 connectivity
Following an older question of mine Maximum connected components after removing 2 vertices. It turns out that for the family of graphs I talk about, we can have 1,2 or 3 connected components after the ...
2
votes
2answers
47 views
Lie group $SU(2)$ is the universal covering group of $SO(3)$.
I need to show that Lie group $SU(2)$ is the universal covering group of $SO(3)$ using the Adjoint representation of $SU(2)$.
But I am stuck at the first step of finding the adjoint representation of ...
0
votes
0answers
9 views
number of connected components of $E(n)$
Let $E(n) := \{f : \Bbb R^n → \Bbb R^n , ||f (x)|| = ||x||\}$ be the group of affine isometries of $\Bbb R^n$.
Prove that $T(n)$ the set of translations $x→ x + y, y ∈ \Bbb R^n$ verifies $T(n) \...
2
votes
1answer
21 views
3-connectivity as a set linear constraints
It turns out connectivity of a graph can be expressed as a set of linear constraints. https://www.researchgate.net/post/How_can_I_ensure_graph_connectivity_using_LP_or_MIP_formulation
Giving a vertex ...
2
votes
1answer
24 views
Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} x H x^{-1}$ is connected
Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} x H x^{-1}$ is connected.
My attempt.
I constructed the function $g_x : H \to S$ such that $g_x(h) = x h x^{-...
0
votes
1answer
19 views
Locally path connected and connected imply path connected
Let $X$ be a connected locally path connected space. I want to show that it is also path connected.
Following a suggestion in this answer, fix $a\in X$ and consider the set
$$U_a = \{x\in X : \text{...
1
vote
1answer
66 views
Locally connected and connected
Let X be locally connected, $A \subset X$ arbitrary. Let $S \subset A$ be connected and open in A. Show $S=U\cap A$ where $U$ is connected and open.
I think that I have solved already but I never ...
0
votes
2answers
31 views
The converse of continuous image of a connected set is connected [duplicate]
I have recently proven that given $f:X\rightarrow Y, f$ continuous, X connected, then Y is connected. I wonder if the converse is true if we consider open mapping. In other words, Given X and Y ...
1
vote
2answers
94 views
There are no other clopen sets in $\mathbb{R}$ except for $\mathbb{R}$ and $\emptyset$
Proof attempt:
Let there be another clopen set $S$ in which is a proper subset of $\mathbb{R}$. Hence, $ S^c \neq \emptyset $.
We can assert the following statements:
No point of $S$ lies in $S^c$...
0
votes
1answer
22 views
Union of conjugates of a connected subgroup of a matrix group is connected.
Let $H$ be any connected subgroup of a matrix group $G$. Show that $S = \bigcup_{x \in G} xHx^{-1}$ is connected.
0
votes
0answers
55 views
Visualising connectedness in topology
Recently started a topology course, I am struggling to visualise connected topological spaces in topology, I am understanding the definition when it comes to clopen subsets and disjointness, ie $[0,1) ...
4
votes
2answers
73 views
Proof - Graph Connectedness
Let $n\ge2$ be an integer. Let $G$ be a graph with $n+1$ vertices and
more than ${n \choose 2}$ edges. Show that $G$ is connected.
Here's my attempted solution to the above problem:
Induction on $...
0
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0answers
54 views
Hausdorff, locally connected and locally compact space reference.
I would like to find a reference to the following proposition:
Let $X$ a Hausdorff, locally compact, locally connected, connected space and $K \subseteq X$ a compact subset. Then, there exists a ...
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0answers
36 views
Topology on $\mathbb R$ with a disconnected subset $A$ where the intersection of seperations of $A$ with $\mathbb{R}-A$ is not $\varnothing$?
$\newcommand{\R}{\mathbb{R}}$
Hello, I was wondering how to find a topology on $\R$ and a disconnected subset $A$ such that every pair of sets, $U$ and $V$, that is a separation of $A$ in $\R$ ...
2
votes
1answer
47 views
$G\subset \mathbb{C}$ simply connected, $f$ holom. on $G$, show every connected comp. of $\{ |f| < c\}$ $(c > 0)$ is simply connected.
I have the tools:
$G$ simply connected $\iff$ every contour in $G$ has winding number $0$ about every point in $\mathbb{C}\setminus G$
$G$ is simply connected $\iff$ every holom. function on $G$ has ...
1
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0answers
15 views
Question regarding removal of a perfect matching in bridgeless graphs
Suppose a connected bridgeless graph has a perfect matching. If we delete the edges forming the perfect matching, then will the graph be still connected?
Will the graph be still bridgeless too?
...
0
votes
2answers
40 views
Connected sub set
Let X be a topological space, and A sub set
how to prove that the following are equivalent:
1) A is connected
2) if A is decomposed into two open sets M and N such that $A=M\cup N$ and $$ (\...
1
vote
1answer
29 views
Example of a Connected Space which is not Locally Connected.
Here is an good example of a connected space which is not locally connected but I am stuck in disproving the local disconnectedness:
Let
$A$ $=$ $\big\{ (x, y)| x \in \mathbb{Q}^c $ and $0 ≤ y ≤ 1$ $...
1
vote
1answer
38 views
prove using the definition of connectedness and paths that $G_3 = (V_1\cup V_2, E_1\cup E_2)$ is a connected graph
Let $ G_1 = (V_1, E_1), G_2 = (V_2, E_2)$ such that...
G1 is connected
G2 is connected
$V_1\cap V_2={v_o}$
I know since the intersection of both vertex sets only contains a single vertex, implying ...
3
votes
1answer
161 views
Representation of $\overline{\mathbb{Q}}$ in One Dimension
Introduction: Below gives an approach to realize the algebraic numbers $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ within a $1$-dimensional compact connected abelian group (solenoid). Essentially a ...
3
votes
3answers
135 views
$X,Y$ not homeomorphic if $X\backslash x$ is disconnected for all $x\in X$ but $Y\backslash y$ is connected for some $y\in Y$
I have seen here on stack exchange (in the comments) a proof along the lines of:
$X\backslash x$ is disconnected for all $x\in X$ but $Y\backslash y$ is connected for some $y\in Y$. Therefore $X$ and ...
0
votes
1answer
37 views
How to Show $(0,\infty ) $ is connected in $R_k$ topology?
I wanted to show that $(0,\infty ) $ is connected in $R_k$ topology.
OPen set in $R_k$ is $(a,b)$ or $(a,b)\setminus K $where $K=\{1/n \mid n\in \mathbb N\}$
As open interval In $(1,\infty ) $ in R ...
0
votes
1answer
34 views
Show that a matrices space is connected
Prove that $2\times2$ complex matrices of determinant 0 and non null trace is connected.
What I did:
My intuition is that such matrices are similar to \begin{pmatrix}
z & 0 \\
0 & 0
\end{...
1
vote
1answer
34 views
Question regarding perfect matchings in a graph
I have a connected, bridgeless graph with 72 vertices of degree 3 and 2 vertices of degree 2. Is there a way that I can prove that this graph has a perfect matching?
From the graph I can see there is ...
1
vote
1answer
63 views
How is the Knaster–Kuratowski fan connected?
If the Knaster–Kuratowski fan is totally disconnected with its dispersion point removed, then how is it connected with its dispersion point in place?
How can the connectedness of the points along the ...
0
votes
1answer
26 views
Example of Set X which is not well ordered such that $X\times [0,1)$ with dictionary ordered is not Linear Continuum
Example of Set X which is not well ordered such that $X\times [0,1)$ with dictionary ordered is not Linear Continuum
I had proved following theorem
Set X which is not well ordered then $X\times [0,...
1
vote
1answer
32 views
Number of connected components in a set
I have a function $H$ with canonical coordinates $(q,p)$ on the cotangent bundle $T^*Q \cong S^1 \times \mathbb{R}$. The function $H$ has 4 critical points and has regular values where ever it is not ...
1
vote
1answer
56 views
$A$ and $B$ connected implies atleast one of $A \cup B$ or $A \cap B$ is connected
Let $A$ and $B$ be connected subsets of a metric space. Prove that at
least one of $A \cup B$ or $A \cap B$ is connected
My two attempts:
Assume neither $A \cup B$ nor $A\cap B$ are connected (...
2
votes
1answer
94 views
What's wrong with the following proof that any interval $(a,b) \subset \mathbb{R}$ is connected?
Suppose not. Let $U, V$ be nonempty disjoint open sets such that $(a,b) = U \cup V$. Let $x \in U, y \in V$, and w.l.o.g. assume that $x < y$. Consider the set $S = \{z \in V | z > x\} \subseteq ...
0
votes
1answer
51 views
Let A be a proper subset of X and B a proper subset of Y. If X and Y are connected, show that $(X\times Y)\setminus (A\times B)$ is connected.
I know that this question is already answered in the site but I did it in a different way but I don't know if it is correct.
My attempt:
Suppose that $(X\times Y) \setminus (A\times B)$ is ...
1
vote
0answers
67 views
How to show that a set is connected
Can someone please explain to me the idea of connectedness and how they relate to these sets? I understand the definition of connectedness, but I am struggling to relate the definition to these ...
1
vote
1answer
41 views
Connected subset of subset is connected in whole space.
Soppose $(X,T)$ is a topological space. Let $Y\subset S\subset X$. Is $Y$ connected in $S$ iff it is connected in $X$.
Possible Proof:
Since $Y$ is connected in S the only closed and open sets in $...
3
votes
1answer
49 views
Is the set of bounded functions connected?
One of my topology homework questions this week says the following:
Consider the space of bounded functions $B[0, 1]$ on the interval $[0, 1]$
with the $d\infty$ metric.
Prove that in the ...
4
votes
0answers
54 views
Number of unique paths on the edges of a grid with wraparound that return to the origin
I was given this problem on the codegolf stackexchange, but I don't know where to begin on how to calculate it, except by creating some brute-force program to do it for me (like almost all existing ...
