
Re: definition of closure in topological space question
Posted:
Nov 17, 2012 11:36 AM


William Elliot wrote: > > On Sat, 17 Nov 2012, Daniel J. Greenhoe wrote: > > > Closure in topological space is defined using at least two different ways in the literature: > > 1. cl(A) is the intersection of all closed sets containing A. > > > 2. cl(A) is the intersection of all neighborhoods containing A, where > > a neighborhood is any set containing an open set (an element of the > > topology). > > Those definitions aren't equivalent. Consider Sorgenfrey's two > point space S = { 0,1 } with the topology { empty set, {0}, S }. > > By 1, cl {0} = S while by 2, cl {0} = {0} isn't even a closed set. > > > Examples of authors who use 1 include Kelley, Munkres, Thron, and McCarty. > > Examples of authors who use 2 include Mendelson and Aliprantis & Burkinshaw. > > > My question is, one definition considered to be more "standard" than the > > other (from my very limited survey, 1 might seem more standard). > > Yes, 1 is the one to be used. 2 is bogus as I showed.
cl(A) 2. should read
cl(A) = {x : for each neighbourhood N of x, N intersect A =/= emptyset}.
A neighbourhood of x is an element of a complete system of neighbourhoods of x, denoted N_x. A complete system of neighbourhoods of x in X satisfies
For all x in X, N_x =/= emptyset; For all x in X and N in N_x, x in N; For all x in X and N in N_x, if M superset N then M in N_x; For all x in X and N, M in N_x, N intersect M in N_x; For all x in X and N in N_x, there is an M in N_x such that M subset N and M in N_y for each y in M.
A subset O of X is open if O is a neighbourhood of each x in O. Thus "neighbourhood spaces" and topological spaces with the usual open set axioms are equivalent.
With those definitions, cl version 1 and cl version 2 are (as one would expect) equivalent. I know nothing about these things, but I just don't want the OP to be confused.
[neighbourhood = neighborhood]
 When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

