1. ## 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???

2. 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.

3. 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...

4. 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.

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