Define a function p r rarr z as follows give r the standard

Question

Q1. Define a function p: R → Z as follows:

Give R the standard topology. Determine the quotient topology on Z determined by p.

Q2. Determine ways to identify the faces of the cube I × I × I so that the resulting identification space is

(a) a 3-torus (i.e. S¹ × S¹ × S¹).

(b) S¹ × RP².

(c) S¹ × K, where K is the Klein bottle.

It may help to draw a cube with letters on the faces (like one you may have played with as a baby) and indicate the identifications by rotating or reflecting the letter. For example, you can indicate that the front face is glued to the back face with a quarter turn by drawing the letter B on the front andon the back. (Don't use quarter turns.)

Q3. A space is totally disconnected if its only connected subspaces are one-points sets. A Hausdorff space is extremely disconnected if the closure of every open set is open.

(a) Show that X is extremely disconnected iff the interior of every closed set is closed iff disjoint open sets have disjoint closures.

(b) Show that Z with the arithmetic progression topology is totally disconnected.

(c) Show that Z with the arithmetic progression topology is NOT extremely disconnected.