\begin{definition}[Definition:Higher-Aleph Complement Topology]
Let $S$ be a set whose cardinality is $\aleph_n$ where $n > 0$.
Let $\tau \subseteq \powerset S$ be the set of subsets of $S$ defined as:
:$\tau = \set {U \subseteq S: \size {\relcomp S U} = \aleph_m: m < n} \cup \set {U \subseteq S: \relcomp S U \text { is finite} } \cup \O$
That is, $\tau$ is the set of subsets of $S$ whose complements relative to $S$ are of a cardinality strictly less than $S$.
Then $\tau$ is an '''$\aleph_m$ complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is an '''$\aleph_m$ complement space'''.
This construction is an extension of the concept of the finite complement topology and the countable complement topology.
\end{definition}