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.