Six Xi with topology Oi (for i = 1,2,3,4,5,6) examples which are topological spaces;

2. Six Ui (for i = 1,2,3,4,5,6) examples that are open subsets of one example of a topological space X with O;

3. Six Ci (for i = 1,2,3,4,5,6) examples that are closed subsets of one example of a topological space X with O;

4. Three subsets Ai (for i = 1,2,3) with the interior int(Ai), the closure Ai, and the boundary ∂Ai from one example of a topological space X with a topology O;

5. Four Bi (for i = 1,2,3,4) examples that are a basis for a topological space(s) (i.e. the topologies can be diﬀerent, just need four bases);

6. For two Bj (j ∈{1,2,3,4} of just above), show that Bj satisﬁes the two properties of a basis (on pg 8 of Hatcher’s notes);

Math 314 Spring 2020: Project 2 page 6 of 6

7. Three functions di : {(x,y) such that x,y ∈ Xi} −→ R that are metric on a set Xi (for i = 1,2,3); here, the sets Xi’s do not have to be diﬀerent just the metrics need to be diﬀerent;

8. For each of the three di on Xi (for i = 1,2,3 of just above), give three Brj(xj) examples where each is an open ball of some radius rj centered at some point xj (for j = 1,2,3)—note: for each i, there are three j’s; so nine total examples are needed;

9. For one of the di on Xi, show the collection of all open balls B = {Br(x) such that x ∈ Xi,r ∈R,r > 0} satisﬁes the two properties of a basis (on pg 8 of Hatcher’s notes);

10. Three fi (for i = 1,2,3) that are continuous function between two topological spaces (i.e. the topological spaces can be diﬀerent for diﬀerent i’s); 11. For one fi (for i ∈ {1,2,3} of just above), show that fi satisﬁes the two lemmas about continuous functions (on pg 12 of Hatcher’s notes); 12. Give one homeomorphism between two topological spaces X with OX and Y with OY where X and Y are diﬀerent sets