Let $X$ denote a topological space. Then each condition in the following list implies the next.
I know that 2 does not imply 1 (see here). I'm also guessing that 3 does not imply 2. Does anyone know of an example?
One example is the co-finite topology on any infinite set $X$. Since any nonempty open set omits at most finitely many points, every subset is compact. However any proper infinite subset is not closed, so there are compact subsets which are not closed. (This space is clearly T1 since all finite subsets are closed by definition.)
Perhaps the simplest example is to let $p$ and $q$ be distinct points not in $\Bbb N$, let $X=\Bbb N\cup\{p,q\}$, make each point of $\Bbb N$ isolated, and give $p$ and $q$ open nbhds as follows. A set $U\subseteq X$ is an open nbhd of $p$ iff $p\in U$ and $\Bbb N\setminus U$ is finite, and $U$ is an open nbhd of $q$ iff $q\in U$ and $\Bbb N\setminus U$ is finite. Essentially we’ve just made $\Bbb N$ a sequence with two limit points, $p$ and $q$. It’s easy to check that $X$ is $T_1$, but $\{p\}\cup\Bbb N$ and $\{q\}\cup\Bbb N$ are compact sets that aren’t closed: they’re both dense in $X$.
This answer has a slightly more complicated example in which all convergent sequences have unique limits. And this answer has another more complicated example.
