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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Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected. Question: Assume…
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Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is…
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Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are there any other metric spaces with this…
Hagen von Eitzen
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Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
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Can a nowhere continuous function have a connected graph?

After noticing that function $f: \mathbb R\rightarrow \mathbb R $ $$ f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right. $$ has a graph that is a connected set, despite the function not being…
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'flimsy' spaces: removing any $n$ points results in disconnectedness

Consider the following property: $\mathbb R$ is a connected space, but $\mathbb R\setminus \{p\}$ is disconnected for every $p\in \mathbb R$. $S^1$ is a connected space and if we remove any point, it is still connected. But if we remove two…
Babelfish
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Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. Surely it's going to be close to apparent that…
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Is the complement of countably many disjoint closed disks path connected?

Let $\{D_n\}_{n=1}^\infty$ be a family of pairwise disjoint closed disks in $\mathbb{R}^2$. Is the complement $$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$ always path connected? Here “disk” means a round, geometric disk. (As shown below,…
Jim Belk
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Topologist's sine curve is not path-connected

Is there a (preferably elementary) proof that the graph of the (discontinuous) function $y$ defined on $[0,1)$ by $$ y(x) =\begin{cases} \sin\left(\dfrac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\\ 0 & \mbox{if $x=0$,}\end{cases}$$ is not path…
persononinternet
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Union of connected subsets is connected if intersection is nonempty

Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected. If $\bigcup\mathscr{F}$ is not connected, then it can be partitioned into two…
PJ Miller
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Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times n}$ with $\det(A) >0$, then there exists a…
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Can path connectedness be defined without using the unit interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational numbers (metric space completion) in order to define any…
Chill2Macht
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Is the complement of an injective continuous map $\mathbb{R}\to \mathbb{R}^2$ with closed image necessarily disconnected?

I am interested in the following Jordan curve theorem-esque question: Suppose that you are given a continuous, injective map $\gamma: \mathbb{R}\to \mathbb{R}^2$ such that the image is a closed subset of $\mathbb{R}^2$. The complement of the…
AnonymousCoward
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Connected, locally connected, path-connected but not locally path-connected subspace of the plane

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in Steen-Seebach, but maybe I overlooked?
R2M
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Why is the "topologist's sine curve" not locally connected?

In Munkres's Topology, it is claimed that "The topologist's sine curve" is not locally connected without further explanation (See Example 3 of Section 25 "Components and Local Connectedness", 2nd edition). The topologist's sine curve: Let $S$…
hengxin
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