be a locally compact metric space. Given two countable indexing sets $\mathcal{A}$ and $\mathbf{T}$, let H and I be two maps from ${\mathbf{T}}\times {\mathcal{A}}\times {\mathbb{R}}_{+}$ into the set of open subsets of $\Omega$ such that

$$\label{H_fn}
 H~:~(t, \alpha, \lambda) \in {\mathbf{T}}\times {\mathcal{A}}\times {\mathbb{R}}_{+} \to H_{\mathbf{t}}(\alpha, \lambda)$$

\
and $$\label{I_fn}
 I~:~ (t, \alpha, \lambda) \in {\mathbf{T}}\times {\mathcal{A}}\times {\mathbb{R}}_{+} \to I_{\mathbf{t}}(\alpha, \lambda)$$\
Furthermore, let $$\label{defH}
 H_{{\mathbf{t}}} (\lambda) := \bigcup_{\alpha \in {\mathcal{A}}} H_{\mathbf{t}}(\alpha, \lambda) \text{ and }  I_{{\mathbf{t}}} (\lambda) := \bigcup_{\alpha \in {\mathcal{A}}} I_{\mathbf{t}}(\alpha, \lambda).$$

Let $\Psi$ denote a set of functions $\psi: {\mathbf{T}}\to {\mathbb{R}}_{+}~:~{\mathbf{t}}\to \psi_{{\mathbf{t}}}$. For $\psi \in \Psi$, consider the limsup sets

$$\label{deflambda}
\Lambda_{H}(\psi) = \limsup_{{\mathbf{t}}\in {\mathbf{T}}} H_{{\mathbf{t}}}(\psi_{{\mathbf{t}}}) \text{ and } \Lambda_{I}(\psi) = \limsup_{{\mathbf{t}}\in {\mathbf{T}}} I_{{\mathbf{t}}}(\psi_{{\mathbf{t}}}).$$

The sets associated with the map $H$ will be called homogeneous sets and those associated with the map $I$, inhomogeneous sets. We now come to two important properties connecting these notions.

The intersection property {#the-intersection-property .unnumbered}
-------------------------

The triple $(H, I, \Psi)$ is said to satisfy the intersection property if, for any $\psi \in \Psi$, there exists $\psi^{*} \in \Psi$ such that, for all but finitely many ${\mathbf{t}}\in {\mathbf{T}}$ and all distinct $\alpha$ and $\alpha'$ in ${\mathcal{A}}$, we have that $$\label{inter}
I_{{\mathbf{t}}}(\alpha, \psi_{{\mathbf{t}}}) \cap  I_{{\mathbf{t}}}(\alpha', 