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

A topological space is connected if it cannot be written as union of two disjoint non-empty open sets. Every topological space can be partitioned into connected components, which are connected subsets which are maximal with respect to inclusion.

Several related properties are studied in topology:

In graph theory, a connected graph is a graph such that there exists a path between any two vertices.

3082 questions
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Is $E= [1,\infty)$ connected in $\mathbb{R}$ with the usual metric

I am in $\mathbb{R}$ with the usual metric and i want to check if $E= [1,\infty)$ which is a subset of $\mathbb{R}$ is connected. I have a theorem that states that if E is connected then for $a,b∈E$ whenever we have $a < x < b $ means that $x∈E$. I…
Edward Colens
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Proof a set is connected

I have the set $$ S=\{(x,y)\in\mathbb{R}^2:(y^2-64)(x^2+y^2-16) = 0 \} $$ I want to show that it is connected and I wasn't sure how to start, the first idea I had was to show the only subsets of S which are open and closed are $\emptyset$ and $x$ as…
ThomasL123
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If every non-trivial subset of a set C has non-empty boundary in C, then C is connected

I came across this proposition in Functional analysis book by Joseph Muscat. I am having problem in applying this to following example. Let's say C = [-1,1] \ {0}, then C is not connected. But I am not able to find a subset with non-empty boundary.…
Prashanth
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Cut points and connected space

Let $p$ a cut point of a connected space $(X, \mathcal{T})$ and suppose $C$ and $D$ form a separation of $X-\{p\}$, i.e $$X-\{p\}=C \cup D$$ Prove that $C \cup \{p\}$ is conected. I have this idea: Since $p$ is a cut point then $X-\{ p \}$ is not…
Alejandra Benítez
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Proof on why regular graphs have an all-one eigenvector?

I came across the theorem : A connected graph is regular if and only if the all-one vector is an eigenvector of A. I am not sure on how to prove every connected regular graph has an all-one eigenvector.
winnie
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Is the open ball connected in normed space?

I know this is not true for metric spaces. Any help for normed space. If I have a normed space $X$, is any ball around a point $x ∈ X$ so connected?
Calmat
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$(A_i)_{i\in E}$ family of connected sets such that $\bigcap\limits_{i\in E} A_i \neq \emptyset$ then $\bigcup\limits_{i \in E} A_i$ is connected

My idea so far is, since $\bigcap\limits_{i\in E} A_i \neq \emptyset$, then exists $p \in\bigcap\limits_{i\in E} A_i$ if $\bigcup\limits_{i \in E} A_i$ is not connected, then exists $A$ and $B$ open sets such that $A \bigcup B = \bigcup\limits_{i…
Silkking
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Connectedness of $\mathbb{C} \setminus E$ given $E$ is connected.

Suppose that $E \subset \mathbb{C}$ is connected. Is it true that $\mathbb{C} \setminus E$ is also connected ? The only few examples I have tell me it seems True but maybe there is a simple counter-example or a simple proof to this. Any help ?
Atmos
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When is the intersection of three cones connected?

I am interested in the set of points $x \in \mathbb{R}^n$ described by $Ax \ge 0$, $Bx \ge c$, and $x > 0$. $A$ and $B$ are $n \times n$ matrices, and $c \in \mathbb{R}^n$. I would like to know under what circumstances this set is…
GMB
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If $A \cup B$ is open and disconnected in $\mathbb{R}^2$, does it follows both $A$ and $B$ are open

Let $A$ and $B$ be two disjoint subset of $\mathbb{R}^2$ such that $A \cup B$ is open and disconnected in $\mathbb{R}^2$. does it follows both $A$ and $B$ are open. If $A$ and $B$ are both open, since $A$ and $B$ are disjoint by hypothesis. then by…
vijayanand
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When is a finite subset of a metric space connected?

When is a finite subset of a metric space connected? My Explanation: Here metric is not given, If i consider discrete metric then set with only one element are connected as the only open set are singletons and whole space X. If it has two or more…
A. Khan
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Prove topological space is connected

Prove : A topological space $X$ is connected if and only if the only continuous functions form $X$ into the discrete space $Y={0,1}$ are the constant functions, $f(x)=0$ or $f(x)=1$.
YEON
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Is $L=\{(x,y)|y=0\}\cup\{(x,y)|x>0, y=\frac{1}{x}\}\subset\mathbb{R}^2$ connected?

My question is: Is $$L=\{(x,y)|y=0\}\cup\{(x,y)|x>0, y=\frac{1}{x}\}\subset\mathbb{R}^2$$ connected or not? I know: A set is connected iff there no exists $A_1,A_2\subset A$ open sets such that $A_1\neq\emptyset,A_2\neq\emptyset$, $A_1\cap A_2…
MathCracky
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S is connected but int(S) is not

I tried to have an example of a set $S\subset\mathbb{R}^2$ such that $S$ is connected but int$(S)$ is not. Can anyone give me an example and prove it? Thanks
Kelan
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Is this subset of the plane path connected?

Inspired by this question I asked myself the question which I am going to describe: Let $\mathbb {I}$ be the set of all irrational numbers. Let $\mathbb {I}^2$ be the Cartesian product of the set $\mathbb {I}$ with itself, in other words, the set of…
A. P.
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