Re: intersections and unions in the definition of a topology

In article <1118377159.423667.118590@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
bode@xxxxxxxxxxxxx wrote:

> ....
> Why are there allowed to be arbitrary unions of open sets in a
> topology, but only finite intersections of open sets? It seems to be a
> rather subtle distinction, so I wonder why is a topology not arbitrary
> unions and arbitrary intersections, or finite unions and finite
> intersections, or even finite unions and arbitrary intersections....


Several people have quite correctly explained that the definition
arose by generalizing special cases which were already familiar. That
was my first thought too.

Another possible explanation concentrates on technical terms. The
case of finite unions and finite intersections has the name "Boolean
algebra of sets" and the case of arbitrary unions and arbitrary
intersections has the name "complete Boolean algebra of sets," so the
separate name "topology" isn't needed for either of those.

But I'm still left with a lingering feeling that the OP hasn't
been fully answered. Looking at the standard modern definition of
topology as above, is there some motivation for it which is intuitively
more direct? Something which makes arbitrary unions and finite
intersections look natural in themselves? I don't know.

Ken Pledger.
.

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