Table of separation properties of various topological spaces I was attempting to make a collection of some of the separation properties of various topological spaces, can anyone tell me which of the properties I got incorrect? Also does anyone know of any source with a collection like this with more topological spaces and more separation properties?
$$\begin{array}{cc|c}
 & \mbox{T1} & \mbox{Hausdorff}&\mbox{Regular} & \mbox{Normal}&\mbox{Separable}\\
\hline
\mathbb{R}&Y & Y & Y&Y&Y\\
\mathbb{R}^n&Y & Y & Y&Y&Y\\
\mbox{indiscrete}&N & N & Y&Y&Y\\
\mbox{discrete}&Y & Y & Y&Y&Y\\
\mbox{Cofinite}& Y& N&N &N&Y \\
\mbox{Cocountable}&Y&N&N&N&Y\\
\mathbb{R}_l&Y &Y &Y &Y&Y\\
\mbox{line w  2 origins} &Y &N &N &N&Y\\
\mbox{ordered  square} & Y&Y &Y &Y&N \\
\mathbb{R}_k &Y &Y &N &N&Y\\
\{0,1\}^A &Y &Y &Y &Y&Y
\end{array}$$
Comment:$\{0,1\}^A$ is the set of all functions from a set $A$ to $\{0,1\}$. $\mathbb{R}$ and $\mathbb{R}^n$ have the standard topology.$\mathbb{R}_l$ is $\mathbb{R}$ with the lower limit topology and $\mathbb{R}_k$ is $\mathbb{R}$ with the $k$ topology. I am not assuming regular/normal necessarily implies $T_1$.
 A: I assume that by ordered square you mean the lexicographically ordered square.
Separability is not a separation property. An uncountable space with the discrete or the co-countable topology is not separable, and $\{0,1\}^A$ is separable if and only if $|A|\le\mathfrak{c}=|\Bbb R|$. (That last result follows from the Hewitt-Marczewski-Pondiczery theorem, mentioned (with links) in this answer.)
Finite spaces with the cofinite topology and countable spaces with the co-countable topology are discrete and therefore regular and normal.
A: As a source of more of such tables the classic book “Counterexamples in Topology” is still very nice (it has such tables in the back for many more such properties), and a modern adaptation (taking that and other sources and making it searchable in a modern way) is $\pi$-base where you can search for a space and find its properties or look for a combination of properties, and find a space for it.
As to your table: separable is not a separation property, despite the name. It’s more of a notion of size IMO.
You cannot say that the cocountable topology is not Hausdorff, because that’s only the case if the underlying set is uncountable. And on a countable or finite set, any topology is separable, so you should specify the exact underlying set in the case of cofinite, discrete, cocountable topologies. Maybe fix the set on $\Bbb R$ for definiteness and contrastive value.
$\{0,1\}^A$ is separable iff $|A|\le |\Bbb R|$, so that needs an extra specification too.
