# 4 Topology Questions : Archimedean Property

I Let (X, T) be a space and A, B C X. Prove

(a) ...

(b) ... Also show that equality does not need to hold.

(e) ....

(d)...... .Also show that equality does not need to hold.

(e) ......

(f) ...

2. Let B {(a, b], b ? R a < b} Show that B is a base for a topology U on R. The topology U is called the upper limit topology.

3. .....

......

Notice that B is the set of all open disks in the plane R x R and 2 is the set of all open rectangles in the plane R x R.

(a) Show that B1 and B2 are bases for a topology on R x R.

(b) Show that B1 and B2 are equivalent bases. In fact, both are bases for the Euclidean topology . on R x R

4. Prove that Q, the set of all rational numbers, is dense in R, ie., ft by showing that any open interval centered at a real number contains rational numbers.

Hint Use the Archimedean property.

See the attached file.

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#### Solution Summary

Topology questions are answered. The solution is detailed and well presented.