Partition, part of partition

"A partition of a set A is a collection of disjoint nonempty subsets of A whose union is all of A." Munkres, Topology, 23.p.

We extend the definition of partition [breaking up into pieces] above to include components of the topological space A.

partition
(i) of a set A
(ii) of a topological space A
<==>
a collection [D] of
(i) disjoint nonempty subsets of A whose union is A, or
(ii) components of A

part of a partition <==> one of the elements [D] of the partition [D]
[which is a
(i) subset of the set A, or
(ii) subspace of the topological space A]

where
A : a set or a topological space
D : the partition of A
D : a part of D