Week 11; 9

Question 1

Define a product topology

Answer 1

Question 2

Prove that the product topology is a topology on X x Y (the 2 spaces)

Answer 2

Question 3

Define projections maps

Answer 3

Question 4

Prove

Answer 4

Question 5

Prove that if X and Y are Hausdorff spaces then X x Y is q Hausdorff space

Answer 5

Question 6

Prove

Answer 6

Question 7

Prove

Answer 7

Question 8

Prove that if X and Y are connected topological spaces then X x Y is connected

Answer 8

Question 9

Prove

Answer 9

Question 10

For a topological space X
and an equivalence relation ~ on X
Denote the equivalence class of x€X
Denote the quotient of X by ~
Define the quotient topology
Denote the quotient space

Answer 10

Question 11

Prove that τ_q is a topology on X/~

Answer 11

Question 12

If V€ τ_q , define q and it’s property?

Answer 12

Question 13

Prove that if a space is compact/connected so too are all of its quotient spaces

Answer 13

7.15:for X and Y topological spaces and f:X->Y continuous. If X is connected=> f(X) is connected

8.18:if X and Y are topological spaces, f:X-> Y is continuous and X is compact => f(X) is compact

Question 14

For X,Y topological spaces, f:X->Y is a quotient map if? And remark?

Answer 14

Question 15

Prove

Answer 15