# Topology Proof

• Mar 12th 2011, 03:04 PM
Dreamer78692
Topology Proof
6. Let X be a non-empty set containing at least two elements, and let a and b be fixed
but different points in X.

(a) Show that the family T = {U ∈ PX | a ∈ U} ∪ {∅} is a topology on X.
(b) Is the family T = {U ∈ PX | a ∈ U or b ∈ U} ∪ {∅} a topology on X?

I don't get this question, what are they asking for, is T the Discrete Topology???(Angry)
• Mar 12th 2011, 03:12 PM
Plato
Quote:

Originally Posted by Dreamer78692
6. Let X be a non-empty set containing at least two elements, and let a and b be fixed but different points in X.
(a) Show that the family T = {U ∈ PX | a ∈ U} ∪ {∅} is a topology on X.

The set $\mathbf{T}$ contains the emptyset and any subset of $\mathbf{X}$ that contains $\mathbf{a}$.
Your task is to show that collection $\mathbf{T}$ is a topology on $\mathbf{X}$.
List the properties of a topology and the check $\mathbf{T}$ for each one.
• Mar 12th 2011, 03:25 PM
Dreamer78692
for (a)
1) ∅ ,X are in T (trivial)
2)for any 2 sets U,V ∈ T U ∩ V has to at least contain a which is an element of T
3) The union of all sets in T will still contain a, therefore it is an element of T

Am I on the right track...(Punch)
• Mar 12th 2011, 03:32 PM
Plato
Quote:

Originally Posted by Dreamer78692
for (a)
1) ∅ ,X are in T (trivial)
2)for any 2 sets U,V ∈ T U ∩ V has to at least contain a which is an element of T
3) The union of all sets in T will still contain a, therefore it is an element of T
Am I on the right track...

Right track yes, but work on the correct language.
Closed with respect to 2) finite intersection and 3) arbitrary union.

• Mar 12th 2011, 03:40 PM
Dreamer78692
for (b)

Let U = {a,c} and V = {b,c} then U ∩ V = {c} which is not an element of T

(Happy)
• Mar 12th 2011, 03:43 PM
Plato
Quote:

Originally Posted by Dreamer78692
for (b)
Let U = {a,c} and V = {b,c} then U ∩ V = {c} which is not an element of T

GOOD! So $\mathbf{T}$ is not a topology because it is not closed under finite intersection.