ONT Re: Topology
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| 1. Topological Spaces
|
| 1.2. Closed Sets
|
| A subset A of a topological space (X, !T!) is 'closed' iff its relative
| complement X~A is open. The complement of the complement of the set A is
| again A, and hence a set is open iff its complement is closed. If !T! is
| the indiscrete topology the complement of X and the complement of the void
| set are the only closed sets; that is, only the void set and X are closed.
| It is always true that the space and the void set are closed as well as open,
| and it may happen, as we have just seen, that these are the only closed sets.
| If !T! is the discrete topology, then every subset is closed and open.
|
| If X is the set of real numbers and !T! the usual topology, then the
| situation is quite different. A 'closed interval' (that is, a set
| of the form {x : a =< x =< b}) is fortunately closed. An open
| interval is not closed and a 'half-open interval' (that is,
| a set of the form {x : a < x =< b} or {x : a =< x < b}
| where a < b) is neither open nor closed. Indeed ...
| the only sets which are both open and closed are
| the space and the void set.
|
| According to the De Morgan formulae, 0.3, the union (intersection) of
| the complements of the members of a family of sets is the complement
| of the intersection (respectively union). Consequently, the union
| of a finite number of closed sets is necessarily closed and the
| intersection of the members of an arbitrary family of closed
| sets is closed. These properties characterize the family
| of closed sets, as the following theorem indicates.
| The simple proof is omitted.
|
| 4. Theorem. Let !F! be a family of sets such that
| the union of a finite subfamily is a member, the
| intersection of an arbitrary non-void subfamily is
| a member, and X = |_|{F : F in !F!} is a member.
| Then !F! is precisely the family of closed sets
| in X relative to the topology consisting of all
| complements of members of !F!.
|
| JLK, Gen Top, page 40.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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