| Revision as of 00:22, 21 January 2002 editTarquin (talk | contribs)14,993 editsmNo edit summary← Previous edit | Revision as of 15:51, 25 February 2002 edit undoConversion script (talk | contribs)10 editsm Automated conversionNext edit → | ||
| Line 1: | Line 1: | ||
| A '''Hausdorff space''' is a ] in which distinct points have disjoint neighbourhoods. Hausdorff spaces are also called T<sub>2</sub> spaces. They are named after ]. | A '''Hausdorff space''' is a ] in which any two distinct points have disjoint neighbourhoods. Hausdorff spaces are also called T<sub>2</sub> spaces. They are named after ]. | ||
| Limits of ] (when they exist) are unique in Hausdorff spaces. | Limits of ] (when they exist) are unique in Hausdorff spaces. | ||
| A ] ''X'' is Hausdorff ] the diagonal {(''x'',''x'') : ''x'' in ''X''} is a closed subspace of the Cartesian product of ''X'' with itself. | A ] ''X'' is Hausdorff ] the diagonal {(''x'',''x'') : ''x'' in ''X''} is a closed subspace of the Cartesian product of ''X'' with itself. | ||
| See also ], ] and ]. | See also ], ] and ]. | ||
| Note: there is a (fairly poor) mathematicians' joke that serves as a reminder of the meaning of this term: in a Hausdorff space, points are "housed off". This pun is so lousy one is almost certain to remember it. | Note: there is a (fairly poor) mathematicians' joke that serves as a reminder of the meaning of this term: in a Hausdorff space, points are "housed off". This pun is so lousy one is almost certain to remember it. | ||
| ] | |||
| ] | |||
Revision as of 15:51, 25 February 2002
A Hausdorff space is a topological space in which any two distinct points have disjoint neighbourhoods. Hausdorff spaces are also called T2 spaces. They are named after Felix Hausdorff.
Limits of sequences (when they exist) are unique in Hausdorff spaces.
A topological space X is Hausdorff iff the diagonal {(x,x) : x in X} is a closed subspace of the Cartesian product of X with itself.
See also topology, compact space and Tychonoff space.
Note: there is a (fairly poor) mathematicians' joke that serves as a reminder of the meaning of this term: in a Hausdorff space, points are "housed off". This pun is so lousy one is almost certain to remember it.