Chapter 2.12: Topological Spaces

Question 1

Definition: Topology

Answer 1

A topology on a set X is a collection T of subsets of X having the properties:

(1) the null set and X are in T
(2) The union of the elements of any subcollection of T is in T
(3) The intersection of the elements of any finite subcollection of T is in T.

A set X for which a topology T has been specified is called a topological space

Question 2

Definition: Open set of a set X

Answer 2

a subset U in X is an open set if it belongs to the collection T, i.e. the topology of X

Question 3

Definition: Discrete topology

Answer 3

If X is any set, the collection of all subsets of X is a topology on X, and is called the discrete topology

Question 4

Definition: Trivial topology, or the indiscrete topology

Answer 4

If X is any set, the collection consisting of X and the null set is a topology on X, and is called the trivial, or indiscrete topology.

Question 5

Definition: Finite complement topology

Answer 5

the collection T_f of subsets U in X such that:

X - U is either finite, or it is all of X.

Question 6

Definitions:

(1) Finer
(2) Strictly finer
(3) Coarser
(4) Strictly coarser
(5) Comparable

Answer 6

Take T and T’

If T is a subset of T’, we say T’ is finer than T, and T is coarser than T’

If T’ properly contains T, then T’ is strictly finer, and T is strictly coarser

If T is a subset of T’ or T’ is a subset of T, then T and T’ are comparable