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Note that while a basis uniquely determines the topology, the same topology can be describd by different possibilities for basis. In other words, if form a basis for, then a basis for the subspace topology on is given by. Namely, for each basis open set, replace it by its intersection with. Given a basis for, we can directly use it to define a basis for the subspace topology on. The description of which subsets are closed completely determines the topology, and the topology completely determines which subsets are open.Īlso, is not uniquely determined by, though, subject to its existence, we can take a minimal, which is the intersection of all possible choices, and is also the closure of within. In other words, is closed in if and only if there exists a closed subset of such that. Ī subset of is closed in if and only if it is the intersection with of a closed subset of. The description of which subsets are open completely determines the topology, and the topology completely determines which subsets are open.Īlso, is not uniquely determined by, though, subject to its existence, we can take a maximal, which is the union of all the possible choices for. In other words, is open in if and only if there exists an open subset of such that. The equivalent formulations are described below:Ī subset of is open in if and only if it is the intersection with of an open subset of. The subspace topology can be defined in many equivalent ways. Thus, subsets of topological spaces are often also called subspaces. Note that induced with this topology is a topological space in its own right. The subspace topology or induced topology or relative topology on can be defined in many equivalent ways. Let be a topological space (viz, a set endowed with a topology ) and be a subset of. View other induced structures on subspaces Definition View a complete list of basic definitions in topology This article describes the induced structure on any subset (subspace) corresponding to a particular structure on a set: the structure of a topological space VIEW: Definitions built on this | Facts about this | Survey articles about this

This article is about a basic definition in topology.