Subtopology

Subtopology, commonly called the subspace topology or trace topology, is the topology on a subset of a topological space that is inherited from the ambient space. If X is a topological space with topology τ and A is a subset of X, the subtopology on A is defined by τA = {U ∩ A : U ∈ τ}. Equivalently, a subset U of A is open in the subtopology on A if and only if there exists an open set V in X with U = V ∩ A.

The subtopology is the coarsest topology on A for which the inclusion map i: A → X is continuous. It is the initial topology with respect to this inclusion. This construction ensures that the open sets of A reflect the openness of sets in X when restricted to A, and therefore properties of X influence the subspace A through restriction.

Examples illustrate the idea. In the real line R with its standard topology, the subtopology on a subset like A = [0,1] consists of sets of the form (a,b) ∩ [0,1], as well as [0,b) and (a,1] where appropriate, obtained by intersecting open sets of R with A. Open and closed subsets of A are precisely the intersections with open and closed subsets of X, respectively.

Many properties are inherited by subspaces, such as T1 and Hausdorff separation. However, compactness is not generally inherited by arbitrary subspaces; a closed subspace of a compact space is compact, but an arbitrary subspace need not be. The subtopology framework is fundamental in studying continuity, functions, and constructions that involve restricting spaces to subsets.