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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.

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When is a union of connected sets connected?

I know a theorem that any union of connected sets is connected if there is a non-empty intersection. I can also think of a counterexample to the converse: take the interior of the unit circle and the circle boundary in $\mathbb{C}$. It seems to me…
A.M.
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Find the connected component of given set.

$Let \;$ = $\{(, ) ∈ ℝ^2: ^2 + ^2 = 1\} \bigcup([−1,1] × {0}) ∪ ({0} × [−1,1]).$ Let$\; _0$ = $\max\{ ∶ < ∞,$ there are$\; $ distinct points$\; _1, … , _ ∈ $ such that $\setminus \{_1, … , _ \}$ is connected$\}$ 1). The value of $_0$ is 2).Let $\{…
gaurav saini
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Is $\left(x, \frac{1}{x}\cdot \cos\left( \frac{1}{x^2}\right)\right)\cup\{(0,0)\}$ a connected set?

Is $$ \left\{\left(x, \frac{1}{x}\cdot \cos\left( \frac{1}{x^2}\right)\right) \mid x \in \mathbb{R} \setminus \{0\}\right\} \cup \{ (0,0)\} \subset \mathbb{R^2}$$ a connected set? I tried proving by contradiction that it is connected , but it…
user560461
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Is R is connected subspace of R2.?

1st I want to know , is R is a subspace of R2? Because R can be written as R×{0} which is a subset of R2.
Subhasish Chowdhury
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There is no non-trivial connected subgroup of $S^1$.

Let us consider the set $S^1=\{z∈C: |z|=1\}$. Show that there is no non-trivial connected subgroup of $S^1$. I know that $S^1$ denotes the unit circle in $R^2$. But how can I show mathematically that there is no non-trivial connected subgroup of…
abcdmath
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Connectedness of $\{(x,y)| x/y \in \mathbb Q\} \cup \{(x,0)|x\ne 0\}$

I am trying to decide if the set $\{(x,y)| x/y \in \mathbb Q\} \cup \{(x,0)|x\ne 0\}$ is connected or disconnected. It is clearly not path-connected because it is impossible to get from line to the other without going through the origin, which is…
user85798
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$\mathbb{R}^{p}\backslash \{0 \}$ is connected for $p \geq 2$

I have to show that $\mathbb{R}^{p}\backslash \{0 \}$ is connected for $p \geq 2$. Is it possible to show this using the property debated on in this article: Union of connected subsets is connected if intersection is nonempty?
simp
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Is the union of two closed disjoint intervals a connected set?

If a connected set cannot be divided in two disjoint non empty open sets, can I say that $[1,2]\cup[3,5]$ is connected?
Roberto Dias Algarte
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When is the intersection of half-space unions path-connected?

I have sets $S_1, \dots, S_m \subset\mathbb{R}^n$ ($m > n$). Each $S_j$ is the union of two half-spaces (a half-space is described by $\{x | a \cdot x \ge b \}$ for some $a \in \mathbb{R}^n$, $b \in \mathbb{R}$). I would like to know if $S_1 \cap…
GMB
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Is this set connected or not?

The set is given: $ \Theta = {(x, y) : (x, y) ∈ Q^c × Q^c}$ where $Q^c$ denotes the set of all irrational numbers. Is this set connected?
Galymbek
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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 prove this or have a proof of this theorem?
user220927
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Connected set is subset of the collection of its accumulation points.

A question from my analysis homework: Let $A \subset M$ be connected and contain more than one point. Show that every point in $A$ is an accumulation point of $A$. This question makes sense to me intuitively because if there existed an $x \in A$…
Kevin Sheng
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is C[a,b] with 1-norm connected

If C[a,b] denotes the set of all real valued continuous functions over [a,b] is it connected w.r.t. the 1-norm ?1-norm of a function f is defined to be integration of f from a to b.
Learnmore
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Simple connected space

I was wondering how to show that $\mathbb{C}\times\mathbb{R}^+$ is simple connected (every closed arc is continuously reducible to a dot). The problem is more how can one write such paths in such space. Can someone help ?
faero
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Locally constant functions in connected set

If $\Omega\in \mathbb{R^n}$ is a connect open set,function $f:\Omega \to \mathbb{R}$ is continious and locally constant. Prove:$f$ is constant function. I try to construct a "path" to connect $x$ and $y$, but in fact $E$ is not path connected，so…
Lagranngekmno4
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