```Date: Nov 17, 2012 3:47 AM
Author: Achimota
Subject: Re: topology definition question

Thank you very much Kaba and Jesse for your help. I appreciate it.If it really does come down to convention, maybe for me it would be best that I just give all 4 of the criteria rather than have to first state what convention I am assuming for the set operations.Thank you again,DanOn Saturday, November 17, 2012 10:33:04 AM UTC+8, Jesse F. Hughes wrote:> Kaba  writes:> > > > > 16.11.2012 23:42, Daniel J. Greenhoe wrote:> > >> It seems the most "common" definition of a topology is that T is a topology on a set X if> > >>    1. empty set is in T and> > >>    2. X is in T and> > >>    3. A and B are in T ==> A intersection B is in T and> > >>    4. {A_i} in T ==> Union A_i is in T.> > >>> > >> But some authors imply that only 3 and 4 are necessary for the definition of a topology. For example, Kelley ("General Topology", 1955, page 37) only uses 3 and 4 and says that these imply X is in T. McCarty ("Topology...", page 87) says 1 and 2 are "completely unneeded".> > >>> > >> My question is, is it really possible to exclude 1 and 2 from the definition such that 3 and 4 alone imply 1 and 2?> > >>> > >> Suppose X:={x,y,z} and T:={ {x},{y},{x,y} }.> > >> Then T satisfies conditions 3 and 4, but yet X is not in T.> > >> So how is it possible to exclude 3 from the definition of a topology?> > >> > > By convention, the intersection of zero number of subsets of X is the > > > whole space X. Similarly, the union of zero number of subsets of X is > > > the empty set.> > > > Yes, but note that one needs to state (3) in terms of closure under all> > finite intersections, rather than closure under binary intersections, in> > order to ensure that (1) follows.> > > > -- > > "I liked the world a lot better over ten years ago. I believed in a> > lot more things. Hell, most people believed in a lot more things.> > Back then the United States was still, well, known as most people used> > to know the United States." -- James S. Harris in a nostalgic mood
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