Week 4: 4

Question 1

Criteria for topology ?
Topological space?
Elements of topology?

Answer 1

Question 2

Discrete topology

Answer 2

Question 3

Trivial topology

Answer 3

Question 4

Metric topology

Answer 4

Question 5

We say that τ is induced by a metric d if

Answer 5

Question 6

Given 2 topologies on one set, denote one being finer than the other

Answer 6

Question 7

Given (X, τ) topological space, if x€ X and A€X then A is a neighbourhood of x if

Answer 7

Question 8

Relate a neighbourhood U of a point x in a metric space to the induced topology

Answer 8

This is an equivalence

Question 9

For a topological space (X, τ) and a subset A€X, denote the subspace topology

Answer 9

Question 10

Given a topological space (X, τ) define the interior of a set A€X

Answer 10

Question 11

Define convergence in a topological space

Answer 11

Question 12

Hausdorff space is given

Answer 12

Question 13

Prove that every metric space is a Hausdorff space

Answer 13

Question 14

Consequence of every metric space being a Hausdorff space

Answer 14

Question 15

Prove uniqueness of limits in a Hausdorff space

Answer 15

Question 16

Given a topological space (X, τ), a point x€X is called a limit point of a set A if

Answer 16

Question 17

Prove

Answer 17

Question 18

Prove

Answer 18