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Proof That the Union of Any Family of Connected Subsets of Any Space Having at Least One Point in Common is Also Connected

Theorem

The union of any family of connected subsets of any spacehaving at least one point in common is also connected.

Proof

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.