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| 1 | Equivalence Relation |
| 1 | Point |
| 1 | Subspace Topology |
| 1 | Topological Space |
| 1 | Property |
| 1 | Collection |
| 1 | Equivalence Class |
| 1 | Relation |
| 1 | Subset |
| 2 | Hereditary Topology |
| 7 | Connected Component |
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Connected Space
A topological space
is said to be connected if there is no pair of nonempty subsets
such that both 
and 
are open
in 
,
and
. If 
is not connected, i.e. if there are sets 
and 
with the above properties, then we say that 
is disconnected.
Every topological space 
can be viewed as a collection
of subspaces
each of which are connected. These subspaces are called the connected components of 
. Slightly more rigorously, we define an equivalence relation
on points
in 
by declaring that 
if there is a connected subset 
of 
such that 
and 
both lie in 
. Then a connected component
of 
is defined to be an equivalence class
under this relation.