Relative topology

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of a subset of a topological space

The system of intersections of all possible open subsets of (i.e. of elements of the topology ) with . The relative topology is often called the induced topology.

A subset of the topological space equipped with the relative topology is called a subspace of . A subspace of a -space is itself a -space, (cf. Separation axiom). A subspace of a metrizable space is itself metrizable. Any Tikhonov space of weight is homeomorphic to a subspace of a Hausdorff compactum of weight (Tikhonov's theorem).



[a1] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 50ff
How to Cite This Entry:
Relative topology. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article