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Topic:
topology definition question
Replies:
5
Last Post:
Nov 17, 2012 11:14 AM




Re: topology definition question
Posted:
Nov 17, 2012 3:47 AM


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, Dan
On 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



