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## 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.