Hausdorff space: ( a sequence converges to at most one limit.)
Example N8.1 (Finite complement topology). Take with the finite complement topology: the opens are and . Let ,, all be distinct. For all and all , the neighborhood contains all but a finite number of the , so .
So in this topology the sequence converges to every point.
Definition N8.2 (Hausdorff). is Hausdorff if for all there exist neighborhoods , which are disjoint from each other: .
Example N8.3. Metric spaces are Hausdorff: take ; then and are disjoint.
Example N8.4 (Finite complement topology, continued). The finite complement topology on is not Hausdorff.
If is Hausdorff and , then the subspace topology on is Hausdorff.
If are Hausdorff, then is Hausdorff.
Theorem N8.5. If is Hausdorff, then every sequence in converges to at most one limit.