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.
Best Answer
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.