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I'm having trouble making sense of certain terminology. So the question asks me to determine whether a finite closed topology on an infinite set is a $T_1$ space. Now before I get into that, I'm having trouble dissecting the statement of a finite closed topology on an infinite set.

So to start, the definition of a finite closed topology on a set $X$ is the closed subsets of $X$ are $X$ itself and all finite subsets of $X$; that is the open sets are the empty set and all subsets of $X$ which have finite complements.

Here is my problem. If this is the definition of the finite complement topology, then what topological space are the open sets on? So a closed set on a topological space is defined to be closed if its complement in $X$ is open in the topology put upon it, but if that is so then how are these open sets even in the finite complement topology if they are open?

Also I was trying to imagine what the finite complement topology would look like on an infinite set I came up with this:

$$X = \{X,\emptyset, \{a_1\},\,\{a_2\},\,\dots\{a_i\},\,\dots\{a_1,a_2\},\dots,\{a_1,a_2,\dots,a_n\}\}$$ and the sets would continue to get larger.

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The "finite closed topology" describes the closed sets. Which gives you an exact definition of the open sets, they are the complements of closed sets. So indeed this is the co-finite topology (or finite complement topology).

A subset $U$ is open if and only if $U=\varnothing$ or $X\setminus U$ is finite.

Imagining the entire topology is a bit tricky, since infinite sets can be very large, and therefore have many finite sets. So it's best to think about this in terms of definitions. There is a definition when a set is open, and when it is closed. Now check if these definitions meet the requirement of being $T_1$.

And for that matter, allow me to remind you that $X$ is a $T_1$ space if and only if every singleton is closed.

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  • $\begingroup$ the open sets are the complements of closed sets. So in this specific topology the "closed" sets are the open sets because they are in the topology, meanwhile the "open" sets are the closed sets because they are the complement to what are the "open" sets on the topological space? I'm really confused... $\endgroup$ – dc3rd May 1 '15 at 20:57
  • $\begingroup$ You can define the topology by giving the open sets; or the closed sets. Note that if all singletons are open, then every set is open. So it is impossible that all finite sets are open, but not all sets are open. $\endgroup$ – Asaf Karagila May 1 '15 at 21:02
  • $\begingroup$ hmmm ok, well if we use the closed definition of a topology, then all of the singleton sets would be members of the finite closed topology, but the complements of those finite singleton sets would be open, but also infinite, which would mean that this is not a $T_1$ space. $\endgroup$ – dc3rd May 1 '15 at 21:34
  • $\begingroup$ I don't get your argument. Finite closed means finite sets are closed, and complement of finite sets are open. Not that finite complements of finite sets are open, because that makes no sense, in an infinite set the complement of size finite set is never finite itself. $\endgroup$ – Asaf Karagila May 1 '15 at 21:38
  • $\begingroup$ what I am saying is that since all of the singleton sets are in the topology and since each member of this topology is closed, that if I take the complement of any singleton set that is a member of the finite closed topology, the complement of that singleton set is infinite. $\endgroup$ – dc3rd May 1 '15 at 21:47

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