Subspace topology

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This article is about a basic definition in topology.
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This article describes the induced structure on any subset (subspace) corresponding to a particular structure on a set: the structure of a topological space
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Definition

Let (X,\tau) be a topological space (viz, a set X endowed with a topology \tau) and A be a subset of X. The subspace topology or induced topology or relative topology on A can be defined in many equivalent ways. Note that A induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.

The subspace topology can be defined in many equivalent ways. The equivalent formulations are described below:

Formulation in terms of ... Details Comment
determining which subsets are open A subset of A is open in A if and only if it is the intersection with A of an open subset of X. In other words, U is open in A if and only if there exists an open subset V of X such that U = V \cap A. The description of which subsets are open completely determines the topology, and the topology completely determines which subsets are open.
Also, V is not uniquely determined by U, though, subject to its existence, we can take a maximal V, which is the union of all the possible choices for V.
determining which subsets are closed A subset of A is closed in A if and only if it is the intersection with A of a closed subset of X. In other words, C is closed in A if and only if there exists a closed subset D of X such that C = D \cap A. The description of which subsets are closed completely determines the topology, and the topology completely determines which subsets are open.
Also, D is not uniquely determined by C, though, subject to its existence, we can take a minimal D, which is the intersection of all possible choices, and is also the closure of C within X.
finding a basis Given a basis for \tau, we can directly use it to define a basis for the subspace topology on A. Namely, for each basis open set, replace it by its intersection with A. In other words, if \{ U_i \}_{i \in I} form a basis for \tau, then a basis for the subspace topology on A is given by \{ U_i \cap A \}_{i \in I}. Note that while a basis uniquely determines the topology, the same topology can be describd by different possibilities for basis. Thus, a priori, it is not clear that different starting choices of basis for X would yield the same topology on A. To show that this is true, we show equivalence with the open subset formulation.
finding a subbasis Given a subbasis for \tau, we can directly use it to define a basis for the subspace topology on A. Namely, for each subbasis open set, replace it by its intersection with A. In other words, if \{ U_i \}_{i \in I} form a subbasis for \tau, then a basis for the subspace topology on A is given by \{ U_i \cap A \}_{i \in I}. Note that while a subbasis uniquely determines the topology, the same topology can be described by different possibilities for basis. Thus, a priori, it is not clear that different starting choices of basis for X would yield the same topology on A. To show that this is true, we show equivalence with the open subset formulation.
in terms of making a map continuous The subspace topology is the coarsest topology that can be endowed on A, for which the inclusion map from A to X is a continuous map.

Equivalence of definitions

Further information: Equivalence of definitions of subspace topology

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 88-89
  • Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 10 (Theorem 6): introduced under the name relative topology