Subspaces of the Real Line

Topology · Axiom Academy

EXAMPLE Subspace Topology on Subsets of ℝ Three worked problems exploring how subspaces inherit topology Find all open sets in with the subspace topology inherited from ℝ. Is the interval [0, 1/2) open in the subspace topology on [0,1]? Show that [0, 1/2) is open in [0,1] with the subspace topology, even though it's not open in ℝ. Which open set in ℝ could we intersect with [0,1] to get (1/4, 1]? Describe the subspace topology on ℤ ⊂ ℝ. Show that it is discrete. Is the subspace topology on ℚ (rationals) ⊂ ℝ discrete?

This is the written version of the interactive lesson above. See the full Topology course.