In topology and related areas of mathematics, a subspace of a topological space ( X , τ ) {\displaystyle (X,\tau )} is a subset S of X which is equipped with a topology induced from that of τ {\displaystyle \tau } called the subspace topology (or the relative topology, inherited topology, induced topology, or trace topology).
Definition Given a topological space ( X , τ ) {\displaystyle (X,\tau )} and a subset S {\displaystyle S} of X {\displaystyle X} , the subspace topology on S {\displaystyle S} is defined by
τ S = { S ∩ U ∣ U ∈ τ } . {\displaystyle \tau _{S}=\lbrace S\cap U\mid U\in \tau \rbrace .}
That is, a subset of S {\displaystyle S} is open in the subspace topology if and only if it is the intersection of S {\displaystyle S} with an open set in ( X , τ ) {\displaystyle (X,\tau )} . If S {\displaystyle S} is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of ( X , τ ) {\displaystyle (X,\tau )} . 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 S {\displaystyle S} of X {\displaystyle X} as the coarsest topology for which the inclusion map
ι : S ↪ X {\displaystyle \iota :S\hookrightarrow X}
is continuous. More generally, suppose ι {\displaystyle \iota } is an injection from a set S {\displaystyle S} to a topological space X {\displaystyle X} . Then the subspace topology on S {\displaystyle S} is defined as the coarsest topology for which ι {\displaystyle \iota } is continuous. The open sets in this topology are precisely the ones of the form ι − 1 ( U ) {\displaystyle \iota ^{-1}(U)} for U {\displaystyle U} open in X {\displaystyle X} . S {\displaystyle S} is then homeomorphic to its image in X {\displaystyle X} (also with the subspace topology) and ι {\displaystyle \iota } is called a topological embedding. A subspace S {\displaystyle S} is called an open subspace if the injection ι {\displaystyle \iota } is an open map, i.e., if the forward image of an open set of S {\displaystyle S} is open in X {\displaystyle X} . Likewise it is called a closed subspace if the injection ι {\displaystyle \iota } 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 S {\displaystyle S} is a subset of X {\displaystyle X} , and ( X , τ ) {\displaystyle (X,\tau )} is a topological space, then the unadorned symbols " S {\displaystyle S} " and " X {\displaystyle X} " can often be used to refer both to S {\displaystyle S} and X {\displaystyle X} considered as two subsets of X {\displaystyle X} , and also to ( S , τ S ) {\displaystyle (S,\tau _{S})} and ( X , τ ) {\displaystyle (X,\tau )} as the topological spaces, related as discussed above. So phrases such as " S {\displaystyle S} an open subspace of X {\displaystyle X} " are used to mean that ( S , τ S ) {\displaystyle (S,\tau _{S})} is an open subspace of ( X , τ ) {\displaystyle (X,\tau )} , in the sense used above; that is: (i) S ∈ τ {\displaystyle S\in \tau } ; and (ii) S {\displaystyle S} is considered to be endowed with the subspace topology.
Examples In the following, R {\displaystyle \mathbb {R} } represents the real numbers with their usual topology.
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