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user153451

  1. If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?

  2. What conditions on $X$ assure a bijectionan injection (if that´s ever possible)?

By infinite topologies, I mean topologies with an infinite number of sets (subsets of $X$) as its elements.

  1. If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?

  2. What conditions on $X$ assure a bijection?

By infinite topologies, I mean topologies with an infinite number of sets (subsets of $X$) as its elements.

  1. If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?

  2. What conditions on $X$ assure an injection (if that´s ever possible)?

By infinite topologies, I mean topologies with an infinite number of sets (subsets of $X$) as its elements.

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user153451
user153451

Questions about existence of injections between infinite sets and the sets of all infinite topologies on them

  1. If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?

  2. What conditions on $X$ assure a bijection?

By infinite topologies, I mean topologies with an infinite number of sets (subsets of $X$) as its elements.