# Topological equivalence

A relation between topological spaces. Two topological spaces and are said to be topologically equivalent if they are homeomorphic, that is, if there exists a homeomorphism from the space onto the space . Topological equivalence is a reflexive, symmetric and transitive binary relation on the class of all topological spaces. This partitions the collection of all topological spaces into pairwise-disjoint topological-equivalence classes. The properties of topological spaces that are preserved under topological equivalence, that is, under arbitrary homeomorphisms, are called topological invariants. Examples: the line and an interval (without end points) are topologically equivalent; the line and a closed interval are not topologically equivalent. Any two triangles are topologically equivalent, but they do not exhaust the topological equivalence class to which they belong — it also contains, for example, all circles. An important extension of the notion of topological equivalence is that of homotopy equivalence (cf. Homotopy type).

#### References

[1] | P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

#### Comments

#### References

[a1] | R. Engelking, "General topology" , Heldermann (1989) |

**How to Cite This Entry:**

Topological equivalence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Topological_equivalence&oldid=14893