# 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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### 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, $f(S)\subset Y$ ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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&...
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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 ...
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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 ...
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### Is there a way to phrase “continuous” in terms of “connected”?

This is something of a pedagogical question. Most of us have scratched our heads at some point as to why the definition of "$f$ is continuous" requires that the preimage of any open set under $f$ is ...
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### Connectedness of the boundary

My question is about the following claim: For $n \geq 2$, let $A\subset \mathbb R^n$ be a non-empty, open, bounded set. Assume $A$ and its complement are connected and $\text{int}(\text{cl}(A)) = A$...
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### Connected topological space such that the removal of any of its points disconnects it into exactly $3$ connected components?

$\mathbb R$ has the property of being a connected space which is divided into $2$ connected components by the removal of any of its points. I'm trying to generalize this property by constructing a ...