Some Different Topologies

When I use a word it means exactly what I want it to mean, nothing more and nothing less.

Humpty Dumpty

from Alice in Wonderland

_________________________________________________________________________________

Let . Define a set $\QTR{Large}{U}$ to be open in $\QTR{Large}{L}$ if:

and is open in .

For example, $\QTR{Large}{(a,b)}$ any $\QTR{Large}{a}$ and any $\QTR{Large}{b}$ .
, and $\QTR{Large}{V}$ is open in .

For example, any $\QTR{Large}{a}$ and any $\QTR{Large}{b}$ . And
, and is open in .

For example, any $\QTR{Large}{a}$ and any $\QTR{Large}{b}$ . And

Show that this forms a topology:

Proof:

$\QTR{Large}{L}$ is open since and is open in
is open in $\QTR{Large}{L}$ since it is open in
That the given subsets are closed under arbitrary unions and finite intersections follows by some simple Boolean algebra . Check that each of the three types of open sets are closed under arbitrary unions and finite intersections. Then check that a union or intersection of one set of type 1. 2. or 3. with another set of type 1. 2. or 3. gives a set of type 1. 2. or 3.

Theorem:

$\QTR{Large}{L}$ is not Hausdorff. In particular, there do not exist open sets $\QTR{Large}{U}$ and $\QTR{Large}{V}$ such that , and

Proof:

Assume then $\QTR{Large}{V}$ is an open set of hence for some .

Assume then for some

But,

___________________________________________________________________

Definition:

Let $\QTR{Large}{T}$ be a topological space and let . We can define a topology on $\QTR{Large}{T}$ which we write as be defining a set $\QTR{Large}{U}$ to be open in if either or $\QTR{Large}{U}$ is open in $\QTR{Large}{T}$ and . We call this the local topology at $\QTR{Large}{a}$ .

One checks that this does define a topology and if $\QTR{Large}{T}$ has at least two points then it is not Hausdorff. All non-empty open sets contain $\QTR{Large}{x}$ .

__________________________________________________________________

Definition:

Let $\QTR{Large}{S}$ be any set. Define $\QTR{Large}{S}$ and all finite subsets to be closed. It is immediate that arbitrary intersections and finite unions of closed sets are closed hence the complements, the cofinite sets, and form a topology. If $\QTR{Large}{S}$ is infinite this is not Hausdorff. We want to look at the cofinite topology on

Lemma:

Suppose we are given such that then $\QTR{Large}{f}$ is continuous.

Proof:

. More generally, the inverse image of finite (closed) sets are finite (closed).

__________________________________________________________________

Theorem:

Let Let be a sequence,

Then if and only if is continuous, having the cofinite topology.

Proof:

This is almost a trivial restatement of the various definitions.

By definition, if for any there exists an $\QTR{Large}{n}$ such that

That is, the inverse image of any open set in is cofinite.

Conversely, suppose the inverse image of any open set in is cofinite in Since is open its inverse image is cofinite that is there exists an $\QTR{Large}{n}$ such that