Subspace topology

In the mathematical branch of topology is meant by the subspace topology (also induced topology, relative topology, trace topology or subspace topology ) the natural structure, which "inherits" a subset of a topological space. The subspace topology is a special initial topology.

Formal definition

It is the basic set of a topological space and a subset. Then the subspace topology is the topology

The open subsets of are exactly the intersections of open subsets of with.

Properties

• The subspace topology on a subset of a topological space is the weakest topology for which the inclusion mapping

• Is an open subset of a topological space, then a subset if and only open in the subspace topology of if as a subset of is open.
• Is a closed subset of a topological space, a subset is then completed exactly in the subspace topology of when is complete as a subset of.
• A continuous map of topological spaces is a monomorphism if and only in the sense of category theory, when applied to the provided with the subspace topology set-theoretic image as a picture is a homeomorphism. In particular monomorphisms are injective.

Examples

• A sheet of paper without margins as a two dimensional object Imagine. In this is not an open set. However, considering the topology with respect to the plane in which the sheet is, then there is an open set.
• The subspace topology is the discrete topology, ie all subsets of are open as subsets of a topological space. For example, the amount of an open subset of, because it is cut open subset of with.