# 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.

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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