subruimtopologie
Subruimtopologie, also called the subspace topology, is a fundamental concept in general topology that describes how to induce a topology on a subset of a topological space from the ambient space. Let (X, τ) be a topological space and let A ⊆ X be any subset. The subspace topology τ_A on A is defined as the collection of all intersections U∩A, where U ∈ τ. Equivalently, a set V ⊆ A is open in the subspace topology if and only if there exists an open set U in X such that V = U ∩ A. This construction guarantees that the inclusion map i : A → X is continuous and that A inherits the “topology” of X restricted to its elements.
A key property of the subspace topology is that it makes every continuous map from a space
Familiar examples include Euclidean space ℝ^n with subsets such as open balls, circles, or arbitrary sets, where