Problem 4.8

Let $\QTR{Large}{T}$ be a Hausdorff space, let then there exists open sets such that:

for all .
for all

Proof (by induction):

For $\QTR{Large}{n=1}$ the Theorem is trivially true. Let

For $\QTR{Large}{n=2}$ this is just the definition of Hausdorff spaces.

Next show that if the Theorem is true $\QTR{Large}{n-1}$ for then it holds for $\QTR{Large}{n}$ .

Choose open sets such that

for all .
for all

Next, for each choose and $\QTR{Large}{W_{i}}$ such that

and for all
for for all

Define MATH and

Note that

for all .
for all since

and for all
for all since