Week 8; 7.2-8.1 + Week 9; 8.2

Question 1

Prove

Answer 1

Question 2

Given 2 points x, y of a topological space X;
Define a path from x to y in X

Answer 2

γ: [0,1] -> X
A continuous function such that γ(0) = x and γ(1) = y

Question 3

Given 2 points x, y of a topological space X;
We say that X is path connected if

Answer 3

There is a continuous function γ for which there is a path from every x, y € X to each other

Question 4

Prove

Answer 4

Question 5

Prove that a path connected set is connected

Answer 5

Question 6

Prove

Answer 6

Question 7

Define an open cover for a topological space X?
Define a sub cover?
Define a finite sub cover?

Answer 7

Question 8

Define a compact topological space

Answer 8

Question 9

Prove

Answer 9

Question 10

Prove that unit interval [0,1] is compact

Answer 10

Question 11

Prove Heine-Borel in 1D?

Answer 11