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,

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