# 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.

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### Connectedness of the boundary of a domain

I've been struggling to prove the following lemma: "Let $\Omega\subset\mathbb{R}^{d}$ be open and bounded with a Lipschitz boundary and such that $\mathbb{R}^{d}\setminus\partial\Omega$ has ...
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### Strongly connected? Binary numbers of length n with bounded hamming weight

I am wondering if the following is true, and if yes how to prove it: Construct a graph where the vertices are binary numbers of length $n$ and have a hamming weight (i.e. digit sum) between $a$ and $b$...
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### Extremally disconnected without Hausdorff

In Theorem T000045 of pi-Base, a proof is given to defend the assertion from Counterexamples in Topology that all Extremally disconnected ($T_2$ where the closure of open is open) spaces are Totally ...
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### Definition of nested partition of the circle

Below is an excerpt from the paper Boundary torsion and convex caps of locally convex surfaces, in which the author defines a so-called nested partition of the circle. I am having a hard time ... 40 views

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### $\Bbb R\setminus \Bbb Q$ the set of irrationals is disconnected [duplicate]

I'm trying to find two non-empty disjoint open set $A, B$ such that $A \cup B = \Bbb R \setminus \Bbb Q$. But can't find
### Let $X=U\cup V$ where $U,V$ open and disjoint, then every connected subset $A$ of $X$ is a subset of $U$ or a subset of $V$.
I am asked to prove that if $X=U\cup V$ where $U,V$ open and disjoint, then every connected subset $A$ of $X$ is a subset of $U$ or a subset of $V$. My way of solution was assuming in contradiction ...