Week 6

Week 6

Define a topological space X with a subspace A. Find and describe a pair of sets that are a separation of A in X. In addition look at your classmates’ topological spaces, subspaces, and separated sets. Discuss the validity of their separation.
For example: Consider ℝu, ℝ with the upper limit topology, whose basis elements are (a,b] where a < b.  Let A = [1,2] so A ⊂ ℝ.  Define U = (0,1] and V = (1,3] and let A' = A ⋂ U and A'' = A ⋂ V.  Then A' and A'' are open in the subspace topology for A since U and V are open in ℝu.  Note that:
A = A' ⋃ A''     A' = {1}      A'' = (1,2]     A' ⋂ A'' = ∅
Thus, A' and A'' are non-empty disjoint open (in the subspace topology for A) sets whose union is A.  Thus, A' and A'' are a separation of A in ℝu and A is a disconnected subspace of ℝu. Note that no separation of A exists in ℝ using the standard topology.