Re: cofinite topology......

In article <dfaq5f$s1m$1@xxxxxxxxxxxxxxxxx>,
mina_world <mina_world@xxxxxxxxxxx> wrote:
>hello.....doctor~
>
>suppose that (X, T) is cofinite topology space.
>(T = {U in X | U^c : finite set or X})
>
>if Y is finite subset of X, int(Y) = empty set.
>
>---------------------------------------------
>is this right ??
>
>i think........
>if X is finite set, this is wrong.
>because,
>if X is finite set, (X,T) is discrete topology space.
>so,
>even if Y is finite set, int(Y) = Y.(because, Y is open set)

That's right. The cofinite topology on a finite set is the discrete
topology. The cofinite topology only becomes "interesting" when the
set X is infinite.

>and i have anothor idea.
>i know that finite set on cofinite topology is closed set.

Also true.

>so, if Y is finite set, all subset of Y is closed set.
>so, int(Y) = empty set.

No. There is no reason why a closed set cannot also be open; the
entire set is both open and closed, as is the empty set. And in more
general spaces, it happens often. Give (0,1) U {2} the induced
topology, and {2} is both open and closed. This does not work.

Instead, if X is infinite, then the only finite set that can be open
is the empty set (for if A is finite, then X-A must be infinite; the
only set with infinite complement which is open is the empty set. So
if A is finite and open, it must be that A is empty).

Thus, if Y is a finite set, then the largest open subset contained in
Y is the empty set, so int(Y)=empty.



--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

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