# topological invariant

A topological invariant^{} of a space $X$ is a property that depends only on the topology^{} of the space, i.e. it is shared by any topological space homeomorphic to $X$. Common examples include compactness (http://planetmath.org/Compact^{}), connectedness (http://planetmath.org/ConnectedSpace), Hausdorffness (http://planetmath.org/T2Space), Euler characteristic^{}, orientability (http://planetmath.org/Orientation2), dimension^{} (http://planetmath.org/InvarianceOfDimension), and like homology, homotopy groups^{}, and K-theory.

Properties of a space depending on an extra structure^{} such as a metric (i.e. volume, curvature, symplectic invariants) typically are not topological invariants, though sometimes there are useful interpretations^{} of topological invariants which seem to depend on extra information like a metric (for example, the Gauss-Bonnet theorem).

Title | topological invariant |
---|---|

Canonical name | TopologicalInvariant |

Date of creation | 2013-03-22 13:42:07 |

Last modified on | 2013-03-22 13:42:07 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 54-00 |