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I am working on some homework for a topology class I am taking, and I was asked to compare a few topologies on a $\mathbb{R}$.

I won't go into detail about the topologies I worked with, but I was able to find that, for given topologies $T_1, T_2, T_3,$ on $\mathbb{R}$, $T_2$ is finer than $T_1$ and $T_3$ is finer than $T_2$, i.e. $T_1 \subset T_2$ and $T_2 \subset T_3$.

My question is, knowing this, can we say that $T_3$ is finer than $T_1$ by transitivity? I have to compare these two, and it would be much easier if I could say that, by transitivity, $T_1 \subset T_3$.

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The answer is "yes". The point is that a topology is a set of sets and set inclusion is transitive.

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