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 (or the relative topology, or the induced topology, or the trace 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). Given a topological space ( X , τ ) {displaystyle (X, au )} and a subset S {displaystyle S} of X {displaystyle X} , the subspace topology on S {displaystyle S} is defined by 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, au )} . 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, au )} . 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 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. 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, au )} 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, au _{S})} and ( X , τ ) {displaystyle (X, au )} 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, au _{S})} is an open subspace of ( X , τ ) {displaystyle (X, au )} , in the sense used below -- that is that: (i) S ∈ τ {displaystyle Sin au } ; and (ii) S {displaystyle S} is considered to be endowed with the subspace topology. In the following, R {displaystyle mathbb {R} } represents the real numbers with their usual topology.