# Tagged Questions

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### Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
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### Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}.$$ Then $U_\varepsilon$ is open but in ...
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### Connected subspaces and homeomorphism

$S_{1} = \{ (x,y)| y=\sin (\frac{1}{x}), 0<x \le 1 \}$ $S_{2} = \{ (x,y)| y=\sin (\frac{1}{x}), -1\le x <0 \}$ Let $S= S_{1} \cup S_{2} \cup \{(0,0)\}$ 1) Is S connected as a subspace ...
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### Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
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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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### Constructing sets from connected sets

I feel that this is probably really obvious, but I don't know how to get started. Is it true that every set in a metric space is the union of connected, pairwise-separated sets? And does this ...