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- Definitions of connected space 3 answers
Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$.
I thought intuitively that this meant that this was like the English dictionary definition of connected- there is no gap in the set.
It is easy to see $(0,1) \cup (2,3)$ is disconnected. Intuitively it seems $[0,1] \cup [2,3]$ is also disconnected- but by the mathematical definition it is connected. I cannot find a $U$ and $V$!
Also the Wikipedia defn is with closed $U$ and $V$ - how can the two be equivalent?