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). Given a topological space and a subset of , the subspace topology on is defined by That is, a subset of is open in the subspace topology if and only if it is the intersection of with an open set in . If 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 of as the coarsest topology for which the inclusion map is continuous. More generally, suppose is an injection from a set to a topological space . Then the subspace topology on is defined as the coarsest topology for which is continuous. The open sets in this topology are precisely the ones of the form for open in . is then homeomorphic to its image in (also with the subspace topology) and is called a topological embedding. A subspace is called an open subspace if the injection is an open map, i.e., if the forward image of an open set of is open in . 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. Thus, whenever is a subset of , and is a topological space, then the unadorned symbols "" and "" can often be used to refer both to and considered as two subsets of , and also to and as the topological spaces, related as discussed above. So phrases such as " an open subspace of " are used to mean that is an open subspace of , in the sense used above; that is: (i) ; and (ii) is considered to be endowed with the subspace topology. In the following, represents the real numbers with their usual topology.