Proof That the Union of Any Family of Connected Subsets of Any Space Having at Least One Point in Common is Also Connected
The union of any family of connected subsets of any spacehaving at least one point in common is also connected.
Letbe any family of connected subsets ofsuch that for eachthere exists withfor eachso thatand
Letbe the spacewith the discreet topology and letbe continuous.
Since eachis connected andis continuous on
Sincefor eachfor all
Henceis not ontoandis connected.