Inherited 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 (or the relative topology, or the induced topology, or the trace topology).
Definition.
Given a topological space formula_1 and a subset formula_2 of formula_3, the subspace topology on formula_2 is defined by
formula_5
That is, a subset of formula_2 is open in the subspace topology if and only if it is the intersection of formula_2 with an open set in formula_1. If formula_2 is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of formula_1. 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 formula_2 of formula_3 as the coarsest topology for which the inclusion map
formula_13
is continuous.
More generally, suppose formula_14 is an injection from a set formula_2 to a topological space formula_3. Then the subspace topology on formula_2 is defined as the coarsest topology for which formula_14 is continuous. The open sets in this topology are precisely the ones of the form formula_19 for formula_20 open in formula_3. formula_2 is then homeomorphic to its image in formula_3 (also with the subspace topology) and formula_14 is called a topological embedding.
A subspace formula_2 is called an open subspace if the injection formula_14 is an open map, i.e., if the forward image of an open set of formula_2 is open in formula_3. Likewise it is called a closed subspace if the injection formula_14 is a closed map.
Terminology.
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. Thus, whenever formula_2 is a subset of formula_3, and formula_1 is a topological space, then the unadorned symbols "formula_2" and "formula_3" can often be used to refer both to formula_2 and formula_3 considered as two subsets of formula_3, and also to formula_38 and formula_39 as the topological spaces, related as discussed above. So phrases such as "formula_2 an open subspace of formula_3" are used to mean that formula_38 is an open subspace of formula_39, in the sense used above; that is: (i) formula_44; and (ii) formula_2 is considered to be endowed with the subspace topology.
Examples.
In the following, formula_46 represents the real numbers with their usual topology.
Properties.
The subspace topology has the following characteristic property. Let formula_61 be a subspace of formula_3 and let formula_63 be the inclusion map. Then for any topological space formula_64 a map formula_65 is continuous if and only if the composite map formula_66 is continuous. 
This property is characteristic in the sense that it can be used to define the subspace topology on formula_61.
We list some further properties of the subspace topology. In the following let formula_2 be a subspace of formula_3.
Preservation of topological properties.
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.
Notes.
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