Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology[1] (or the relative topology,[1] or the induced topology,[1] or the trace topology).

[2] Given a topological space

, the subspace topology on

is open in the subspace topology if and only if it is the intersection of

is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of

Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset

as the coarsest topology for which the inclusion map is continuous.

is an injection from a set

is defined as the coarsest topology for which

The open sets in this topology are precisely the ones of the form

is then homeomorphic to its image in

is called a topological embedding.

is called an open subspace if the injection

is an open map, i.e., if the forward image of an open set of

Likewise it is called a closed subspace if the injection

is a closed map.

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions.

is a topological space, then the unadorned symbols "

considered as two subsets of

as the topological spaces, related as discussed above.

is considered to be endowed with the subspace topology.

represents the real numbers with their usual topology.

The subspace topology has the following characteristic property.

be the inclusion map.

is continuous if and only if the composite map

This property is characteristic in the sense that it can be used to define the subspace topology on

We list some further properties of the subspace topology.

If a topological space having some topological property implies its subspaces have that property, then we say the property is hereditary.

If only closed subspaces must share the property we call it weakly hereditary.