Posted: Wed Sep 29, 2004 10:08 pm    Post subject: Point Set Topology

Show that if X is Lindeloff and Y is compact then X x Y is Lindeoff
Thanks

Posted: Wed Sep 29, 2004 10:38 pm    Post subject:

what is definition of Lindef…?

Posted: Sun Nov 07, 2004 11:14 am    Post subject:

A topological space is said to be Lindeloff
if any open covering of the space has a countable subcover.

Posted: Sun Nov 07, 2004 2:04 pm    Post subject:

The proof is essential the same as the proof that the product of two compact spaces is compact. Firstly, we notice two simple fact. Fact 1:

The space is Lindeloff if for any opening covering ,

there exists a countable collection of open sets which
covers the space and each of is contained in some . Fact 2:

In order to prove that the space XY is Lindeloff, it is

sufficient to consider opening coverings of the form

where and

are open sets of X and Y respectively.[/dollar]

Posted: Sun Nov 07, 2004 2:46 pm    Post subject:

We can start our proof now.
Let be an opening covering of the product space.

Fix , for each , there exists an element in ,

denoted by which contains the point .
is an opening covering of , so by compactness
of , we may select a finite subcover, say .
Denote the which corresponds to by
and define
Now is an opening covering of , by assumption, there exists a countable subcover, denoted by

.

Finally, the family and is a countable open covering of the product space, and each element of this family is a subset of some element of . Q.E.D.

$[/dollar]