Week 5; 5-5.3 (accidentally did up to 6)

Question 1

Given 2 metric spaces (X, ρ) (Y, d) define a map f: X -> Y continuous at α € X

Answer 1

Question 2

Sequential characterisation of continuity in a map from one metric space to another

Answer 2

Question 3

Prove sequential characterisation of continuity for a map from one metric space to another

Answer 3

Question 4

Prove

Answer 4

Question 5

For metric space (X, d) with fixed element x_0 € X, define continuous map

Answer 5

Question 6

For metric space (X, d) with fixed subset A € X, define continuous map

Answer 6

Question 7

Prove

Answer 7

Question 8

Define the direct product of spaces (X_1, d_1) and (X_2, d_2)

Answer 8

Question 9

Define the pre image of A under f

Answer 9

Question 10

Define a map continuous at α for 2 metric spaces in terms of balls

Answer 10

Question 11

Inverse image characterisation of continuity

Answer 11

Question 12

Proof of inverse image characterisation of continuity

Answer 12

Question 13

Prove

Answer 13

Question 14

Define a continuous map between topological spaces

Answer 14

Question 15

Define an isometry for metric spaces

Answer 15

Question 16

Define a homeomorphism on topological spaces

Answer 16