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.

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