ONT Re: Topology
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| 1. Topological Spaces
|
| 1.5. Interior and Boundary (cont.)
|
| The last statement of the foregoing theorem deserves a little
| further consideration. For convenience, let us denote the
| relative complement X~A by A’. Then A’’, the complement
| of the complement of A, is again A (we sometimes say ’
| is an operator of period two). The preceding result
| can then be stated as A°’ = A’˜, and, it follows,
| taking complements, that A° = A’˜’. Thus the
| interior of A is the complement of the closure
| of the complement of A. If A is replaced by
| its complement it follows that A˜ = A’°’, so
| that the closure of a set is the complement
| of the interior of the complement. *
|
| * An amusing and instructive problem suggests itself. For a given subset
| A of a topological space, how many different sets can be constructed by
| successive applications, in any order, of closure, complementation, and
| interior? From the remarks in the above paragraph and the fact that
| A˜˜ = A˜, this reduces to: How many distinct sets may be formed from
| a single set A, by alternative applications of complementation and the
| closure operator? The surprising answer is given in problem 1.E.
|
| JLK, Gen Top, pages 44-45.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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