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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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2 Answers2

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$[1,2]$ and $[3,5]$ are open subsets of $X:=[1,2]\cup[3,5]$. Indeed, e.g. $[1,2] = (0,3) \cap X$ and $[3,5] = (2,6) \cap X$.

Note that a set is the union of two disjoint non-empty open sets iff it's the union of two disjoint non-empty closed sets (in either case, such sets are clopen in the space).

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When we say that a subset $A$ of a topological space $X$ is connected, we mean that $A$ is a connected topological space with the subspace topology inherited from $X$. In the case of $A=[1, 2]\cup[3, 5]$, both $[1, 2]$ and $[3, 5]$ are closed and open sets in the subspace topology inherited from $\mathbb{R}$. So $A$ is a disjoint union of two nonempty open sets and therefore $A$ is disconnected.

Prajwal Kansakar
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  • I've read that a set $A_1$ is open when all of its elements are centers of an open ball subset of $A_1$. Considering this definition, how can $A_1:=[1,2]$ be open? I ask this because an open interval centered in 2 will never be subset of $A_1$. – Roberto Dias Algarte Apr 16 '17 at 14:26
  • An open interval of length, say $1/2$ centered around $2$ would be ${x\in A:|x-2|<1/2}=(3/2, 2]$ which is a subset of $A$. – Prajwal Kansakar Apr 16 '17 at 15:59