Re: about function space

From: William Elliot (marsh_at_privacy.net)
Date: 10/21/04 

Date: Thu, 21 Oct 2004 03:39:37 -0700

From: Krunk <geosup1@yahoo.it>
Newsgroups: sci.math
Subject: about function space

>In topology, a subbase (or subbasis) for a topological space G with
>topology T is a subcollection B of T such that every open set in T
>can be written as a union of finite intersections of elements of B.
>We say that the subbase generates the topology T, and that T is
>generated by B.

B subbase for a topology on S when B subset P(S) and
        { /\F | F finite subset B }
is _base_ for a topology on S.

/\ intersection; \/ union; \ relative complement; - set difference

>The compact open topology on C(X,Y) is generated by the
>subbase comprising all sets of the form
> {f in C(X,Y) : f(K) subset V}
>where K subset U is compact and V subset Y is open.

>My ask is:
>By definition, every open set in the compact open topology on C(X,Y)
>can be written as a union of finite intersections of elements of B.

No, see my comment above for correct definition of subbase.

>Well, What does open sets like in compact open topology on C(X,Y) ?

An open set is a union of finite intersections of subbase sets. 

----

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