The attractors here form a series $A_n$ such that $A_n=\{x:f^n(x)=x, f^i(x)\neq x~\forall ~i<n,x\in \mathbb R\}$, where $f(x)=3.9x(1-x)$. One can easily see that these satisfy the first criteria.

As for the second criteria, this need not be the case. From this¹ paper (which specifically discusses one dimensional maps):

Let $x^*$ be a fixed point of $f$. Then the basin of attraction (or the stable set) $W^s(x^*)$ is defined as $$W^s(x^*)=\{x:\lim\limits_{n\to\infty}f^n(x)=x^*\}$$

Note the lack of requirement for the basin to be a neighborhood.

If you go a bit further on the Wikipedia page, we see:

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that $B(A)$ be a neighborhood

With this in mind, we can make it satisfy the second criteria.

This is pretty simple: we find the set of numbers $B_n={x:f(x)=a_n ~\forall~ a_n \in A_n}$. This need not be the entire basin, we just need a portion to prove that $A_n$ is an attractor. Of course, we need to prove that $\exists b_n\in B_n$ that is not an element of $A_n$. This can be done by noticing that $f(x)=a$ has 2 roots $\forall~a\in(0,1)$. Since the roots of $f(x)=a_n$ cannot overlap for different $a_n$ (otherwise the two $a_n$s would be equal), we get $2n$ distinct values for which $f(x)\in\A_n$. Since $A_n$ has only $n$ elements, we have some extra numbers which can be considered to be part of the basin of attraction.

^{1. Elaydi, S., & Sacker, R. J. (2003). The basin of attraction of one-dimensional maps.}

The attractors here form a series $A_n$ such that $A_n=\big\{x:f^n(x)=x, f^i(x)\neq x~\forall ~i<n,x\in \mathbb R\big\}$, where $f(x)=3.9x(1-x)$. One can easily see that these satisfy the first criteria.

As for the second criteria, this need not be the case. From this¹ paper (which specifically discusses one dimensional maps):

Let $x^*$ be a fixed point of $f$. Then the basin of attraction (or the stable set) $W^s(x^*)$ is defined as $$W^s(x^*)=\{x:\lim\limits_{n\to\infty}f^n(x)=x^*\}$$

Note the lack of requirement for the basin to be a neighborhood.

If you go a bit further on the Wikipedia page, we see:

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that $B(A)$ be a neighborhood

With this in mind, we can make it satisfy the second criteria.

This is pretty simple: we find the set of numbers $B_n={x:f(x)=a_n ~\forall~ a_n \in A_n}$. This need not be the entire basin, we just need a portion to prove that $A_n$ is an attractor. Of course, we need to prove that $\exists b_n\in B_n$ that is not an element of $A_n$. This can be done by noticing that $f(x)=a$ has 2 roots $\forall~a\in(0,1)$. Since the roots of $f(x)=a_n$ cannot overlap for different $a_n$ (otherwise the two $a_n$'s would be equal), we get $2n$ distinct values for which $f(x)\in A_n$. Since $A_n$ has only $n$ elements, we have some extra numbers which can be considered to be part of the basin of attraction.

^{1. Elaydi, S., & Sacker, R. J. (2003). The basin of attraction of one-dimensional maps.}