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I came up with this question when I drew diagrams to represent the relations among special topological space. I am aware that the answer might be trivial, but I haven’t figure out a way to do it yet.

Given two topological properties $P_1$ and $P_2$ such that $P_1<P_2$($P_1$ implies $P_2$, but does not equal to $P_2$. An example would be “$T_0<T_1$”), is there a topological property $P$ such that $P_1<P<P_2$?(“$<$” is defined as above) Can we define a topology using the class of topological properties?

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The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.

So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $\mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $\mathbb{R}$).

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