let A be Subset of GX; :: thesis: ( A is a_component implies A is closed )
A1: A c= Cl A by PRE_TOPC:18;
assume A2: A is a_component ; :: thesis: A is closed
then Cl A is connected by Th19;
then A = Cl A by A2, A1;
hence A is closed by PRE_TOPC:22; :: thesis: verum