Noun
subspace topology (plural subspace topologies)
(topology) The topology of a subset S of a topological space X which is obtained by considering any subset of S to be an open set if it corresponds to the intersection of S with some open set of X.
A subset of a topological space is said to be connected if it is connected under its subspace topology. Source: Internet
However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R × need not be continuous with respect to the subspace topology. Source: Internet
If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X. More generally, given a set S, specifying the set of continuous functions : into all topological spaces X defines a topology. Source: Internet
More explicitly, an injective continuous map between topological spaces and is a topological embedding if yields a homeomorphism between and (where carries the subspace topology inherited from ). Source: Internet
This construction is dual to the construction of the subspace topology. Source: Internet