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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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「subspace topology」の詳細全文を読む スポンサード リンク
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