Are any of the topologies metrizable?

[tylerc1991]

Homework Statement

Given a set X = {1,2,3} and the following topologies:

T1={{1}, empty set, X}
T2={{1}, {2}, {1,2}, empty set, X}
T3={{1,2}, {2,3}, {2}, empty set, X}
T4={{1}, {2}, {1,2}, {2,3}, empty set, X}
T5={{1,2}, empty set, X}
T6={{1}, {2,3}, empty set, X}
T7={{1}, {1,2}, empty set, X}

Are any of the topologies metrizable? Explain.

The Attempt at a Solution

So I started by looking for the subsets that could be realized as the union of open balls. I was picturing a 'discrete metric' type scenario where each point is itself an open ball of r<1. In that case it looks like T5 contains the only subset that can be expressed as the union of open balls (the open balls being a ball around 1 and 2). Any help is definitely appreciated.

 

[micromass]

Do you know what the T1-axiom is? And do you know that every metrizable space is T1?

 

[tylerc1991]

OK that definitely led me on the right track in determining what type of spaces the various topologies are, but now I have a new question: is a T0-space metrizable? (or a T1-space or T2-space for that matter?) Thanks again for your help.

 

[micromass]

No, a T0-space is in general not metrizable. However, the converse holds: if a space is metrizable, then it is T0. So if you find that your space is not T0, then it cannot be metrizable.

The exact same thing holds with T1, T2, T3, T4, T5 and T6...

So, since none of your spaces is T1, they cannot be metrizable.

In fact: a finite space is metrizable if and only if it is discrete...