Topological equivalence

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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).


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



[a1] R. Engelking, "General topology" , Heldermann (1989)
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Topological equivalence. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article