 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2011/4/ii/solution

S

cannot be expressed as

S = A \cup B

with

A, B

disjoint nonempty relatively open sets. Equivalently, the only subsets simultaneously relatively open and closed are

\emptyset

and

S

. A decomposition into such

A, B

is a separation of the topological space.

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