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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.

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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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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 $n =1$, ...
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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 ...
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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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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?
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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1answer
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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 ...
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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 ...
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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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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 ...
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1answer
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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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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 ...
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2answers
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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 ...
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0answers
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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) \...
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1answer
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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 ...
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1answer
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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^{-...
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1answer
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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{...
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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 ...
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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 ...
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2answers
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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 $S^c$...
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1answer
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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.
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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) ...
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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 $...
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0answers
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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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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 $\R$ ...
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1answer
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$G\subset \mathbb{C}$ 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 ...
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0answers
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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? ...
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2answers
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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 $$ (\...
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1answer
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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 $0 ≤ y ≤ 1$ $...
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1answer
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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 ...
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1answer
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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 ...
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3answers
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$X,Y$ 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 ...
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1answer
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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 ...
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1answer
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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{...
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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 ...
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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 ...
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1answer
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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,...
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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 ...
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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 (...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...