Relations between different definitions of polar topologies

Question

Let $\langle X, Y \rangle$ be a separated dual pair of (real) vector spaces. A non-empty family $\mathcal{A}$ of non-empty subsets of $Y$ is called polar if

1. every $A \in \mathcal{A}$ is bounded (w.r.t. to the duality)
2. every point of $y$ is contained in some $A \in \mathcal{A}$: $\ \bigcup \{ A \mid A \in \mathcal{A} \} = Y$
3. $\mathcal{A}$ is directed by inclusion: $\ \forall B, C \in \mathcal{A} \, \exists A \in \mathcal{A}: \, B \cup C \subseteq A$
4. $\mathcal{A}$ is closed under scalar multiplication: $\ \forall \lambda \in \mathbb{R} \, \forall A \in \mathcal{A}: \, \lambda A \in \mathcal{A}$.

This is the definition as in Wikipedia and many other books in the literature. A polar family $\mathcal{A}$ on $Y$ induces a locally convex topology (the polar topology) on $X$, namely the topology of uniform convergence on $\mathcal{A}$. Due to property 2. this topology is Hausdorff. The weak topology $\sigma(X, Y)$ is the weakest polar topology ($\mathcal{A}$ consists of all finite sets) and the strong topology $\beta(X,Y)$ the strongest polar topology ($\mathcal{A}$ consists of all bounded sets).

Wilansky, "Modern Methods of Topological Vector Spaces" uses a slightly more general definition which uses 1., 3. and

4'. $\mathcal{A}$ is directed under doubling: $\ \forall A \in \mathcal{A} \, \exists B \in \mathcal{A}: 2A \subseteq B$.

So, Wilansky omits property 2. and weakens property 4. He then considers a locally convex topology on $X$ generated by the set of all polars $A^o$ where $A \in \mathcal{A}$. Let us call this a W-polar topology. He shows that this topology is just the topology of uniform convergence on $\mathcal{A}$. Every W-polar topology is weaker than $\beta(X,Y)$. A W-polar topology is then called admissible if it is also finer than $\sigma(X,Y)$. He then shows that a W-polar topology is admissible iff 2'. holds where

2'. $\bigcup \{ A^{oo} \mid A \in \mathcal{A} \} = Y$

is a weakening of property 2.

Question 1: Am I right to say that admissible topologies are just the usual polar topologies as defined above?

Question 2: I was surprised to see Exercise 8-5-105:

"Give an example of a separated nonadmissible W-polar topology."

So the topology in question must be weaker than $\sigma(X,Y)$ or equivalently not satisfying 2'., but still be Hausdorff and W-polar. Do you have some ideas?